Matching centroids by a projective transformation

Given two subsets of Rd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^d$$\end{document}, when does there exist a projective transformation that maps them to two sets with a common centroid? When is this transformation unique modulo affine transformations? We study these questions for 0- and d-dimensional sets, obtaining several existence and uniqueness results as well as examples of non-existence or non-uniqueness. If both sets have dimension 0, then the problem is related to the analytic center of a polytope and to polarity with respect to an algebraic set. If one set is a single point, and the other is a convex body, then it is equivalent by polar duality to the existence and uniqueness of the Santaló point. For a finite point set versus a ball, it generalizes the Möbius centering of edge-circumscribed convex polytopes and is related to the conformal barycenter of Douady-Earle. If both sets are d-dimensional, then we are led to define the Santaló point of a pair of convex bodies. We prove that the Santaló point of a pair exists and is unique, if one of the bodies is contained within the other and has Hilbert diameter less than a dimension-depending constant. The bound is sharp and is obtained by a box inside a cross-polytope.


The setup
This work arose from the following question: Given a convex polytope P in R d and a point q inside P, does there exist a projective transformation ϕ such that ϕ(q) is the centroid of the vertices of ϕ(P)?
For example, if P is a d-simplex, then as ϕ one can take the projective transformation that fixes the vertices of P and maps q to the centroid of P.
The question can be generalized as follows: Question 1 Given two subsets K 1 , K 2 ⊂ R d , does there exist a projective transformation ϕ : RP d → RP d such that the centroids of ϕ(K 1 ) and ϕ(K 2 ) coincide?
The centroid γ (K ) can be defined (by the usual integral formula) for any subset K ⊂ R d that has a positive finite k-Hausdorff measure for some 0 ≤ k ≤ d, see e. g. [9]. Thus, any of the sets K i in Question 1 may be, say, a convex polytope or the k-skeleton of a convex polytope.
If dim K = 0 or dim K = d, then the centroid of K is affinely covariant: γ (ψ(K )) = ψ(γ (K )) ∀ψ ∈ Aff(d) (1) so that we can quotient out affine transformations when searching for ϕ. Since dim Proj(d) − dim Aff(d) = d, the "number of equations" becomes equal to the "number of variables", and the following question poses itself.
For example, if K 1 is the vertex set of a simplex, and K 2 is a single point, then ϕ is unique in the above sense.
For dim(K ) / ∈ {0, d} the centroid is in general not affinely covariant. Indeed, it is wellknown that the centroid of the boundary of a triangle coincides with the centroid of its vertices if and only if the triangle is regular. As the centroid of the vertices is affinely covariant, the centroid of the boundary is not. See [9] for centroids of skeleta of simplices in higher dimensions.
Let us discuss some restrictions we are going to impose on K i and ϕ. First, if dim K i = d, then K i is assumed to be compact and equal to the closure of its interior. The latter is not really a restriction, since replacing K by the closure of int K doesn't change the centroid.
Second, we will always assume one set to be contained in the convex hull of the other: K 2 ⊂ conv K 1 . Although this looks quite restrictive, it leaves enough room for non-trivial results. For an idea of what can be done in the case when the convex hulls are incomparable, see Sect. 7.
Third, in order for the centroid of ϕ(K ) to be defined, no point of K may be sent to infinity for dim K = 0 and vol(ϕ(K ) ∩ R d ) < ∞ must hold for dim K = d. The following restriction (together with compactness of K ) guarantees both.
Non-admissible projective transformations are more difficult to handle; besides, admissible transformations will often suffice. If we allow a projective transformation to send to infinity a hyperplane that separates the points p 1 , . . . , p n , then we can lose the uniqueness, see Proposition 3.3.
The requirement that K is equal to the closure of its interior forbids K to have "antennas". It turns out, we also need to forbid "horns" in order to ensure the existence of a suitable projective transformation. Definition 1.2 A d-dimensional compact subset K ⊂ R d is called cusp-free, if for every x ∈ K ∩ ∂ conv K there is a d-simplex contained in K with a vertex at x.
Examples of cusp-free sets are: pure d-dimensional polyhedra (finite unions of convex d-dimensional polyhedra); d-submanifolds of R d with smooth boundary; d-submanifolds with corners.

Making a given point to the centroid of a set
Here we present our results in the case when one of the sets is a point.
Theorem 1 Let K = {p 1 , . . . , p n } ⊂ R d be a finite set of points affinely spanning R d , and let q ∈ int conv(K ) be a point in the interior of their convex hull. Then there exists a projective transformation ϕ : RP d → RP d , admissible with respect to K , such that ϕ( p 1 ) + · · · + ϕ( p n ) If ψ is any other admissible projective transformation with γ (ψ(K )) = ψ(q), then ϕ • ψ −1 is an affine transformation.
A projective transformation modulo post-composition with affine ones is uniquely determined by the hyperplane ⊂ R d that it sends to infinity. Associate to every the point q that becomes the centroid of { p i } after is sent to infinity. Then, by Theorem 1, the hyperplanes disjoint from conv{ p 1 , . . . , p n } are in one-to-one correspondence with the points inside the convex hull. This correspondence is related to the polarity with respect to an algebraic set. Namely, let A be the union of the hyperplanes dual to { p i }; then the dual of q is the polar with respect to A of the dual of . See Proposition 3.2 for more details.
On the other hand, there is a relation to the analytic center of a polytope and the Karmarkar's algorithm, [2]. Theorem 2 Let K ⊂ R d be a compact cusp-free d-dimensional set, and q ∈ int conv(K ). Then there exists a projective transformation ϕ : RP d → RP d , admissible with respect to K , such that γ (ϕ(K )) = ϕ(q) For any other admissible projective transformation ψ with this property, the composition ϕ • ψ −1 is affine.
Since projective transformations can be represented by central projections (Sect. 2.2), Theorem 2 can be reformulated as existence and uniqueness of a hyperplane section of a cone through a given point having this point as the centroid. Representing projective transformations as composition of two polarities, we can relate Theorem 2 in the case of convex K to the Santaló point: the hyperplane that must be sent to infinity is dual to the Santalo point of the dual of K . See Theorems 2B and 2C in Sect. 4.1.
If the point q lies sufficiently close to a sufficiently sharp cusp of K , then there is no projective transformation making q to the centroid of K . See Example 4.8.

One of the sets is finite
Here we present the results in the case when one of the sets is finite but consists of more than one point. . . , q m } be such that K 2 ⊂ int conv(K 1 ). Then there exists a projective transformation ϕ, admissible with respect to K 1 (and hence with respect to K 2 ), such that γ (ϕ(K 1 )) = γ (ϕ(K 2 )).
In general, ϕ is not unique, even up to post-composition with affine transformations.
For non-uniqueness, see Examples 3.5 and 3.6.

Theorem 4
Let K ⊂ R d be a compact cusp-free d-dimensional set, and let p 1 , . . . , p n ∈ conv K be such that every support hyperplane of K contains less than n d+1 of the points p 1 , . . . , p n . Then there exists a projective transformation ϕ such that In general, ϕ is not unique, even modulo affine transformations.
For the sharpness of the assumptions and for non-uniqueness, see Examples 5.2 and 5.3. Interestingly enough, the assumptions leading to existence become obsolete, and the transformation turns out to be unique, if K is a ball. Since the image of a ball under an admissible projective transformation is an ellipsoid, and the ellipsoid can be mapped back to the ball by an affine transformation, the following theorem is equivalent to the existence and uniqueness of a projective transformation matching the centroids of a ball and of a finite set.
be the unit ball centered at the origin, and let p 1 , . . . , p n ∈ B d be a finite set of points, n ≥ 3. Then there exists a projective transformation fixing B d such that The transformation ϕ is unique up to post-composition with an orthogonal transformation.
This result generalizes centering via Möbius transformations [13] used for unique representation of polyhedral types. There is also a relation to the conformal barycenter [5], see Remark 5.8.

Two convex bodies
Here we present the results for the case when both K 1 and K 2 are d-dimensional. In order to get some uniqueness results, we need to assume that K 1 and K 2 are convex. The uniqueness can be guaranteed if one of the bodies lies "deep inside" the other.
where a, b ∈ ∂ K 1 are points collinear with p and q, and cr( p, q; a, b) The maximum Hilbert width of K 2 with respect to K 1 is defined as where α is a (d − 2)-dimensional affine subspace, and i , m i ⊃ α are support hyperplanes to K i .
It follows immediately from definition that where K • denotes the polar dual of K (one may take the polar duals with respect to any point lying in the interior of both K 1 and K 2 ). See Fig. 2, where L i = K • i . Note that for K 1 = B d the number 1 2 log | cr(m 2 , 2 ; 1 , m 1 )| is the hyperbolic distance between 2 and m 2 , with B d viewed as the Cayley-Klein model of the hyperbolic space. Thus, the maximum hyperbolic width is defined as the maximum distance between support hyperplanes.
In general, ϕ is not unique modulo affine transformations. It is unique, if The bound (3) is sharp. It is achieved for a cross-polytope K 1 and a rectangular parallelepiped K 2 inside it, provided that at least one of the sides of the parallelepiped is sufficiently long, see Fig. 3. The discussion in Sect. 2.3 justifies the following definition.
Here L • y denotes the polar dual of L with respect to the unit sphere centered at y.
By going to the polar duals of K 1 and K 2 , one derives from Theorem 6 criteria for existence and uniqueness of a Santaló point of a pair. Corollary 1.5 Let L 1 , L 2 ⊂ R d be two convex bodies such that L 2 ⊂ int L 1 . Then the pair (L 1 , L 2 ) has at least one Santaló point.
If the Hilbert diameter of L 2 with respect to L 1 satisfies

Plan of the paper and acknowledgments
In Sect. 2 we discuss right cosets of the affine group in the projective group and describe several ways to choose a representative from each coset. Section 3 deals with the case of dim K 1 = dim K 2 = 0, that is with finite point sets. Theorems 1 and 3 are proved here. The solution is found as a critical point of a concave functional (12), respectively of the difference of two such functionals.
In Sect. 4, Theorem 2 is proved and interpreted in the contexts of minimizing the volume of a cone section and of the Santaló point. Here the convex functional (18) associated with a convex body is introduced.
Section 5 deals with the case dim K 1 = d, dim K 2 = 0 and proves Theorems 4 and 5. Section 6 deals with the case of two d-dimensional sets and proves Theorems 6. Finally, in Sect. 7 we pose some questions for future research.
The author wishes to thank Arnau Padrol, Raman Sanyal, Boris Springborn, and Günter Ziegler for useful discussions.

Choosing a hyperplane to be sent to infinity
Affine transformations (or affinities) of R d are maps of the form x → Ax + b, where A ∈ GL(d). Identify R d with a subset of the projective space: by associating x ∈ R d with the equivalence class of (1, x) ∈ R d+1 . Then projective transformations of RP d restricted to R d have the form In particular, the group Aff(d) of affinities is a subgroup of the group Proj(d) of projectivities.
or two representatives (y doing the same as −y) of the form Proof Two projectivities belong to the same right coset of Aff(d) if and only if they send to infinity the same hyperplane. Any hyperplane that does not pass through the origin has equation x, y + 1 = 0 for a unique y, and is therefore sent to infinity by a map of the form (6). In particular, for y = 0 the hyperplane at infinity is sent to itself. Any hyperplane through the origin is sent to infinity by a map of the form (7).
We may always assume 0 ∈ conv(K i ) for i = 1, 2. Then none of the maps (7) is admissible in the sense of Definition 1.1, and the map (6) is admissible if and only if y This allows us to reformulate Questions 1 and 2 as follows.
Reformulation A For a set K ⊂ R d containing the origin in the interior of the convex hull, when does there exist y ∈ R d such that γ (ϕ y (K )) = 0? For two sets K 1 , K 2 ⊂ R d containing the origin in their convex hulls, when does there Under what assumptions is y unique?

Cone sections
Here we give a further reformulation of Questions 1 and 2. We will show this to be equivalent to Reformulation A. Let us introduce some notation. For every set X ⊂ R d+1 define the conical hull over X as

Reformulation B
For x ∈ R d denotex Sections of C by hyperplanes not passing through 0 are central projections ofK , and thus images of K under projective transformations. It suffices to show that central projections modulo dilations are the same as projectivities modulo affinities. For every vector y ∈ R d denote Denote by ρ y : R d → H y the central projection and by π :

Proposition 2.2 The map (6) is a composition of a central and a parallel projection:
And since π(x) = x, we have ϕ y = π • ρ y .
The hyperplane (10) contains e 0 , and every hyperplane through e 0 not passing through 0 is H y for some y ∈ R d . This establishes a bijection between the images ϕ y (K ) and sections H y ∩ C and shows that Reformulation B is equivalent to Reformulation A.

Composition of two polarities
Similarly to (8), define the polar dual of L with respect to a point y: Here is a reformulation of Questions 1 and 2 in the case of a convex body and a point and in the case of two convex bodies.
Reformulation C For a convex body L ⊂ R d , when does there exist a point y ∈ int L such that the polar dual of L with respect to y has centroid at y?
For two convex bodies L 1 , L 2 ⊂ R d , when does there exist a point y ∈ int L 1 ∩ int L 2 such that the centroids of the polar duals of L 1 and L 2 with respect to y coincide?
Under what assumptions is y unique?
Again, we justify this by relating polarity with variable center to the map ϕ y from (6).

Proposition 2.3 For every d-dimensional convex body K ⊂ R d and every point y
, and the proposition follows.

Remark 2.4 The Santaló point s(L) of a convex body L is the point such that
vol(L • s(L) ) ≤ vol(L • y ) ∀y ∈ int L. The existence and the uniqueness of the Santaló point was proved in [11], see also [12, p. 546]. In loc. cit. it was also proved that the property y = γ (L • y ) is characteristic for the Santaló point of L, which implies a positive answer to Questions 1 and 2 in the case of a convex body and a point.
In the case of two convex bodies L 1 and L 2 in Reformulation C the point y can be called the Santaló point of a pair of convex bodies.

One point versus several
Here we prove Theorem 1 using Reformulation A from Sect. 2.1. Without loss of generality we may assume q = 0. Since ϕ y (0) = 0 for all y, it follows from Proposition 2.1 that Theorem 1 is equivalent to the following.
Theorem 1A Let p 1 , . . . , p n ∈ R d be such that 0 ∈ int P, where P = conv{ p 1 , . . . , p n }. Then there exists a unique y ∈ int P • such that The proof is based on the fact that the left hand side of (11) is the gradient of a strictly concave function. Define Then one easily computes The function F is strictly concave, which basically follows from the strict concavity of log x on R, as F is a sum of logarithms of affine functions whose linear parts span (R d ) * . More exactly, the second derivative of F is As p i are affinely spanning R d , they are also linearly spanning it, so that all scalar products vanish only if u = 0.
Finally, as y tends to ∂ P • , some of 1 + p i , y tend to 0, and their logarithms tend to −∞. On the other hand, since P • is bounded, all summands in (12) are bounded from above. Thus F tends to −∞ as y → ∂ P • .
A strictly concave function on a convex set which tends to −∞ as the argument approaches the boundary has a unique critical point, the global maximum. This proves the theorem.

Polarity with respect to a union of hyperplanes
Fix the points p 1 , . . . , p n affinely spanning R d . Theorem 1 says that there is a bijection between the points in the interior of conv{ p i } and the hyperplanes disjoint from conv{ p i }: a hyperplane corresponds to the point q ∈ int conv{ p i } that becomes the centroid of { p i } when is sent to infinity by a projective transformation. (In particular, the hyperplane at infinity corresponds to the actual centroid of { p i }.) In this section we will relate this correspondence to the polarity with respect to an algebraic set.
For a homogeneous degree n polynomial f on a vector space V denote by the same letter f the corresponding n-linear symmetric form: By duality, hyperplanes in V correspond to one-dimensional subspaces of V * . Thus the polarity determines a map P(V \ ker n−1 f ) → P(V * ). Similarly, for a polynomial f on V * one gets a map Proposition 3.2 Let 1 , . . . , n , and m be 1-dimensional subspaces of a vector space V , and let ρ ∈ V * be a linear functional on V such that ker ρ doesn't contain any of i , m. Then the following conditions are equivalent.
Here we denote ker( is an algebraic set of degree n. Proof Choose arbitrary points p i ∈ i and q ∈ m different from the origin. By assumption, , and q ρ(q) (13) Thus the first condition is equivalent to On the other hand, for the polynomial Thus the polar of ρ is the following hyperplane in V * : Thus the second condition is equivalent to which is equivalent to (14) and thus to the first condition.
In the Reformulation B of our problem we have not a collection of lines, but a collection of rays. This means that each of the points (13) is assumed to lie in a specified half of the corresponding line, i. e. the number ρ( p i ), respectively ρ(q) must have a specified sign. In other words, the point [ρ] ∈ P(V * ) must lie in a specified component of the complement P(V * ) \ n i=1 ker p i . By counting the components one can determine the number of classes of projective transformations that send q to the centroid of the images of { p i }.
For the points (13) (withp i andq instead of p i and q) in the affine hyperplane {ρ(x) = 1} there is a unique class of admissible projective transformations making q to the centroid of { p i } if and only ifq ρ(q) lies in the interior of the convex hull ofp i ρ(p i ) . The latter condition says that ρ belongs to a component of P(V * )\ n i=1 kerp i that is disjoint from kerq. Besides, any two functionals from the same component give rise to the same class of projective transformations. There are n(n−1) 2 + 1 components in total, and kerq intersects n of them, which leads to the number in the proposition.

Several points versus several points
Proof of Theorem 3 Without loss of generality 0 ∈ conv(K 1 ), so that the hyperplane sent to infinity by ϕ cannot pass through the origin. By Proposition 2.1 we may look for ϕ among the maps of the form (6). The condition γ (ϕ(K 1 )) = γ (ϕ(K 2 )) then says Similar to the proof of Theorem 1A, the solutions of (15) are the critical points of the function defined in the interior of (conv , so that the function tends to −∞ as y tends to the boundary of conv(K 1 ) • , and hence attains its maximum. The point of minimum yields a desired projective transformation. For non-uniqueness, see Examples 3.5 and 3.6.

Remark 3.4
The sum (16) may tend to −∞ under less restrictive assumptions than conv K 2 ⊂ int conv K 1 . For example, it does so when K 1 consists of the vertices of a triangle in R 2 , and K 2 of three points on the sides of the triangle.
On the other hand, if K 1 is the vertex set of a tetrahedron in R 3 , and K 2 consists of three points on one edge and two points on the opposite edge, then there is no projective transformation that matches the centroids of K 1 and K 2 (the centroid of a tetrahedron lies in the plane parallel to a pair of opposite edges and equidistant from them). In particular, in this case the sum (16) does not tend to −∞ near the boundary of the domain.
One may argue that the above example only works because K 1 is not in convex position. For d = 1 this is actually true: if K 1 = {−1, 1} and K 2 ⊂ (−1, 1), then the solution is unique. In higher dimensions this does not help, as the following example shows. The reason for the failure is that even if K 1 is in convex position, its projections are not. Example 3.6 Take the following subsets of the plane: Again, both have centroid at the origin. Their images under the projective transformation (x, y) → x 2−x , y 2−x both have centroids at ( 1 6 , 0).

Cone sections and the Santaló point
is contained in an open halfspace whose boundary hyperplane passes through the origin. A closed pointed cone possesses bounded sections by affine hyperplanes. We will consider only those sections that intersect each ray of the cone, and call them complete. A pointed cone is the conical hull (see equation (9)) of any of its complete sections.

Criticality of the volume
Let C ⊂ R d+1 be a pointed (d + 1)-dimensional cone. For every hyperplane H such that C ∩ H is compact and intersects all rays of C, denote the bounded component of C \ H by C H .

Proposition 4.2 A hyperplane section H ∩ C of a cone C has centroid at q if and only if H is a critical point of the function
on the set of all hyperplanes through q.
A formal proof of this proposition will be given in the next subsection. Here we sketch a geometric argument without providing all the details.

Remark 4.3
In 1931, Tricomi [14] and Guido Ascoli [1] showed that for every point inside a convex body there exists a hyperplane section that has this point as a centroid. Tricomi dealt only with dimension 3 using the "hairy ball theorem". Ascoli used a variational approach based on Proposition 4.2. They also characterized those non-convex bodies, for which the centroid of a section depends continuously on the hyperplane, making both approaches applicable. For more details see [4, §2, Section 8] that also deals with a beautiful related object, Dupin's "floating body".
Filliman [7] studied critical sections of polytopes and gave their characterization in the case of a simplex.
Our plan now is to show that for cones with cusp-free sections the function (17) is coercive, which implies the existence of a critical point, and then to prove some sort of convexity of (17) to ensure the uniqueness of the critical point.

Logarithmic convexity of the volume and a proof of Theorem 2B
For a non-zero vector y ∈ R d+1 denote The section H y ∩ C is compact and intersects each ray of C if and only if y ∈ int C * , where Note that if C is (d +1)-dimensional, closed and pointed, then C * is also (d +1)-dimensional, closed and pointed.
For every y ∈ int C * denote That is, C y is the bounded part of C cut off by the hyperplane H y . Theorem 2B is proved by using variational properties of the function The following arguments are a slight modification of [8] and [6].

Lemma 4.4 We have
because C y is a pyramid over C ∩ H y with the altitude y −1 .
The second and the third integrals are computed similarly. Take into account that γ (C ∩ In particular, the function (18) equals Proof Let us compute the derivative of F in the direction u ∈ R d+1 :

Lemma 4.6 If K is cusp-free, then the value F(y) tends to +∞ as y tends to a point in
Proof Let y 0 ∈ ∂C * \ {0}. Then there exists x 0 ∈ C such that x 0 , y 0 = 0 and x 0 = 0. Clearly, x 0 ∈ ∂C. Thus, by assumption of Theorem 2 there are vectors x 1 , . . . , x d ∈ R d+1 such that their positive hull = { d i=0 λ i x i | λ i ≥ 0} is contained in C. Then we have for some positive constant. Hence As y tends to y 0 , the scalar product x 0 , y tends to 0 while other scalar products remain bounded below by a positive constant (some of them may also tend to +∞). Hence F(y) → +∞.

Proposition 4.7 The function F is strictly convex.
Proof We have Due to the functional arithmetic-quadratic mean inequality (which is the L 2 Cauchy-Schwarz inequality for functions f and 1) we have

Proof of Theorem 2B
The hyperplane H y passes through the point q if and only if the hyperplane H q passes through y. The section H y ∩ C is bounded if and only if y ∈ int C * . Thus the hyperplane sections of C coming into question are Restrict the function F defined in (18) to C * ∩ H q . By Lemma 4.5 we have grad F| C * ∩H q (y) = γ (C ∩ H y ) − q (This is the projection of grad F to H q ; one may also evoke Lagrange multipliers.) Thus q is the centroid of C ∩ H y if and only if y is a critical point of F| C * ∩H q . Lemma 4.6 implies that F attains a minimum on C * ∩ H q , which shows the existence part of Theorem 2B. The uniqueness follows from Proposition 4.7, similarly to the proof of Theorem 1A.  3 } and q = (0, 0). We claim that none of the maps 0 . The set ϕ a,0 (K ) is symmetric with respect to the x-axis, and it can be shown that for b = 0 the image under ϕ 0,b of an x-symmetric set has its centroid outside the x-axis. It follows that the only candidates for ϕ are the maps For ϕ a,0 to be admissible, we have to assume |a| > 1.
A direct computation shows that the centroid of ϕ a,0 (K ) always has a negative xcoordinate. In particular, in the limit case a = −1 we have Therefore finding the minimum of vol d+1 (C y ) is equivalent to finding the minimum over all y of the volume of the polar dual of L with respect to y ∈ int L. This is the second characterization of the Santaló point of a convex body L, the first having been given in Theorem 2C. The maximum of the product vol(L) vol(L • ) over all origin-symmetric convex bodies is achieved when L is an ellipsoid. This is the Blaschke-Santaló inequality [3,10,11]. The minimum of vol(L) vol(L • ) is not known, but is conjectured to be achieved when L is a cube or cross-polytope or, more generally, Hanner polytopes (Mahler conjecture).

Existence and non-uniqueness in the general case
Here we prove Theorem 4.
By combining the functionals (12) and (18) we see that the classes of projective transformations matching the barycenters are in a 1-to-1 correspondence with the critical points of the function If p i ∈ int conv K for all i, then, for a cusp-free K , the integral tends to +∞ as y tends to ∂C\{0}, while the sum remains bounded. This implies the existence if all p i lie in the interior of conv K . If some of them lie on the boundary, then we need a more delicate argument.

Lemma 5.1
If every support hyperplane of K contains less than n d+1 of the points p 1 , . . . , p n , then the function F tends to +∞ as y tends to a point in ∂C * \ {0}.
Proof Let y → y 0 ∈ ∂C * \ {0}. As in the proof of Lemma 4.6, choose a point x 0 ∈ C such that x 0 , y 0 = 0. Then we have Now, we have p i , y 0 ≥ 0. If for all i this inequality is strict, then all log p i , y remain bounded as y → y 0 , so that F(y) → +∞. Let p 1 , y 0 = 0 and p i , y 0 > 0 for i = 1. As p 1 ∈ conv(C), there exist x 1 , . . . , x k ∈ C such that This implies x i , y 0 = 0 for all i. It follows that On the other hand, Collecting all terms we get Due to λ i ≤ λ and n > d + 1, all coefficients before the logarithms are negative. Hence F(y) → +∞. The restriction on the number of points lying on the boundary of the convex hull is necessary for the existence, as the following example shows.

Example 5.2
Let K be the union of two d-simplices whose intersection is a (d − 1)-face of both (a bipyramid), and let p i , i = 1, . . . , d be the vertices of one of the simplices. Then the centroid of { p 1 , . . . , p n } coincides with the centroid of the corresponding simplex and therefore is different from the centroid of K . No projective transformation can help.
Alternatively, take three points on one edge of the tetrahedron and two points on the opposite edge. The centroid of the points lies on a plane parallel to both edges that divides the distance between them in proportion 2 : 3. The centroid of the tetrahedron lies on a plane equidistant from both edges.
In the following example there are no points on the boundary, but the transformation ϕ is still not unique.

Example 5.3
Let K ⊂ R 2 be the square with vertices (±1, 0), (0, ±1), and Both K and { p 1 , p 2 } have centroid at the origin. The images of both sets under a projective non-affine transformation (x, y) → x 3−x , y 3−x have centroids at 1 12 , 0 . Compare this with Examples 3.5 and 3.6. The coincidences are not accidental.

Centering of points inside a sphere
Here we prove Theorem 5. As usual, we form the difference of functions It follows that where R i is the ray generated by p i . Therefore we have to show that the function F has a unique, up to scaling, critical point. Note also that F(λy) = F(y), so that it suffices to consider the restriction of F to any subset that is represented in all equivalence classes y ∼ λy. Two convenient choices are {y | y 0 = 1} and {y | y 1,d = 1}.

Lemma 5.5 The function F tends to +∞ as y tends to a point in ∂C \ {0}.
Proof Let y → z with z 1,d = 0. Then − log y 1,d → +∞. If p i , z = 0 for all i, then the other summands in (20) remain bounded, and the sum tends to +∞.
If there is an i such that p i , z 1,d = 0, then p i = z, so that only the i-th summand under the sum sign in (20) tends to −∞. We then have This already implies the existence of a critical point of F. For uniqueness we would like to use convexity, but the following example shows that F is not always convex.

Example 5.6
For d ≥ 3 put p i = e 0 + ae 1 + b i e 2 and consider the restriction of F(y) to the line y = e 0 + te 1 . There we have For a = 0.95 this function is not convex.
The following trick helps.

Lemma 5.7
The function F is geodesically strictly convex with respect to the hyperbolic metric on int C/{x ∼ λx}.
Proof When restricted to { y 1,d = 1}, the function F has the form Every geodesic is represented by a hyperplane section of the hyperboloid { y 2 1,d = 1}, and has a unit speed parametrization of the form y(t) = q cosh t + r sinh t where q 2 1,d = 1, r 2 1,d = −1, q, r 1,d = 0. Let us study the restrictions of the i-th summand in (21) to geodesics.
If p i 2 1,d > 0, then on any geodesic it is possible to choose q and r so that p i , r 1,d = 0. We get log p i , y 1,d = log cosh t + const (22) which is strictly convex. If p i 2 1,d = 0 and the geodesic doesn't havep i as a limit point, then one can do the same. If p i 2 1,d = 0 and the geodesic hasp i as a limit point, then for any parametrization we have p i , q 1,d = − p i , r 1,d , so that Thus along such a geodesic the function is linear. The only possibility for the sum (21) to be linear along a geodesic (and thus non-strictly convex) is that all points { p i } have p i 2 1,d = 0 and lie on that geodesic. This is only possible for n ≤ 2.
Proof of Theorem 5 By Lemma 5.5, the function F has a critical point inside the cone C. By Lemma 5.7, this critical point is unique up to scaling, because otherwise the restriction of F to a geodesic would have two different critical points, which contradicts the strict geodesic convexity of F.

Remark 5.8
This argument generalizes that of Springborn [13], who considers only points on the sphere. In this case, the function F is the sum of hyperbolic distances to horospheres centered at the given points. For a point p inside the ball, the term log p, y equals log cosh dist( p , y), where dist is the hyperbolic distance, and p =p/ p 1,d .
In the case when all p i lie on the sphere, the critical point of the function F is the so called conformal barycenter of { p i }. In [5], the conformal barycenter was defined for non-atomic measures on the sphere, and the construction for discrete measures was indicated.

Existence and non-uniqueness
We approach Theorem 6 in Reformulation B: take cones C 1 and C 2 over K 1 and K 2 , respectively, and see under what conditions there is an affine hyperplane H such that C 1 ∩ H and C 2 ∩ H have a common centroid.
Following Sect. 4.3, introduce the functions has, according to Lemma 4.5, the gradient Thus the following lemma holds. The existence of a critical point follows by the usual argument.
As next we give an example where the projective transformation matching the centroids is not unique. Example 6.2 Take a unit disk and the following rectangle inside it: for some a < 1 √ 5 . Both K 1 and K 2 have centroid at the origin. On the other hand, the projective transformation Two bodies whose centroids can be matched in different ways maps the disk K 1 to itself, and the rectangle K 2 to a trapezoid that, as a tedious computation shows, also has centroid at the origin. See Fig. 11. An example of this sort is possible whenever the rectangle has a side which is longer than √ 3.

Uniqueness for K 2 deep inside K 1
We will consider the restriction of F to the section of int C * 1 by the hyperplane y 0 = 1. Since F(λy) = F(y), every critical point of this restriction is a critical point of F. We are not able to prove that F| y 0 =1 is convex under assumption (3), but we can prove that it is strictly convex at every critical point. Since the indices of critical points of a function defined on a ball and tending to +∞ near boundary sum up to 0, this implies that the critical point is unique. Lemma 6.3 Let y ∈ R d+1 be such that y 0 = 1. Then for every vector u ∈ R d+1 with u 0 = 0 we have By Proposition 2.2, ϕ y (K i ) is the image of C i ∩ H y under parallel projection along e 0 . Therefore Also, γ (ϕ y (K i )) differs from γ (C i ∩ H y ) by a multiple of e 0 . It follows that for u 0 = 0 we can replace C i ∩ H y by ϕ y (K i ) in the formula for D 2 F i . Further, for any K ⊂ R d and γ = γ (K ) we have By substituting this into the last equation we obtain and the lemma follows.

Lemma 6.4 Let K ⊂ R d be a convex body, and u ∈ R d be a non-zero vector. Then we have
is the width of K in the direction of u. The lower bound is achieved for a symmetric double cone over any (d − 1)-dimensional body, the upper bound is achieved for the cylinder over any (d − 1)-dimensional body.
Proof If each section of K orthogonal to u is replaced by a (d − 1)-ball of the same radius, then the body remains convex and preserves its volume and moment of inertia in the direction u. Thus, without loss of generality, K is a "rotation body" with axis u.
We will use the fact that moving mass away from the centroid increases the moment of inertia, and moving towards decreases the moment.
By the above principle, the Steiner symmetrization with respect to u ⊥ preserves the volume but decreases the moment. The resulting body is symmetric with respect to a hyperplane orthogonal to u, and it is possible to move more mass towards the centroid by replacing each of the symmetric halves by a cone of the same volume with the base on the hyperplane of symmetry. The two steps are illustrated on Fig. 12. The convex profiles stand for the radii of the sections orthogonal to u; equally colored regions correspond to sets of equal d-volume.
For a double cone of width 2a we have which yields the lower bound in the theorem.
In order to prove the upper bound, first replace K with a truncated cone K 1 whose parts on either sides from the hyperplane through the centroid of K have the same volumes as the corresponding parts of K . One can go from K to K 1 by moving mass away from the centroid, see Fig. 13, therefore K 1 has a bigger moment of inertia.
It turned out unexpectedly hard to prove directly that the cylinder maximizes the moment of inertia among all truncated cones of a fixed volume, therefore we will continue to move Fig. 13 Maximizing the moment of inertia for fixed width and volume mass. We replace K 1 by a union K 2 of a cylynder and a truncated cone as shown on Fig. 13. A direct computation shows that the requirement vol(K 2 \ K 1 ) = vol(K 1 \ K 2 ) leads to a convex K 2 (the radius of the cone decreasing as on Fig. 13). Also, the section of K 2 \ K 1 by a hyperplane orthogonal to u has a smaller volume as the section of K 1 \ K 2 at the same distance from the centroid of K 1 . This allows to map K 1 \ K 2 to K 2 \ K 1 so that the mass is moved away from the centroid of K 1 . Thus I u (K 2 ) ≥ I u (K 1 ).
The body K 2 can be replaced by a truncated cone K 3 with a bigger moment, as it was done at the first step. By iterating the procedure, we obtain a sequence of bodies converging to a cylinder. This implies that the cylinder maximizes the moment of inertia for given volume and width.
The ratio for the cylinder of width 2a equals Lemma 6.5 Let a 1 < a 2 < b 2 < b 1 . Then for every κ ∈ (0, 1) we have Proof For a fixed cross-ratio, the maximum of b 2 − a 2 is achieved when the segments are concentric, that is x = −y. We have x ) 2 and the lemma follows. Proof The function F attains its minimum in D, therefore it has at least one critical point. Due to D 2 F(y) > 0 at all critical points, critical points are isolated. Due to F(y) → +∞ as y → ∂ D, the critical values form a discrete subset of R, and in particular can be ordered: Let a ∈ (F(y k ), F(y k+1 )). By the Morse theory, the set F −1 (−∞, a) (−∞, a). This contradiction shows that the critical point is unique.

Proof of uniqueness in Theorem 6
We are considering the restriction of F to int(C * 1 ∩ {y 0 = 1}) = int K • 1 . From Sect. 6.1 we know that F(y) → +∞ as y → ∂ K • 1 . Let us show that under assumption (3) the quadratic form D 2 F is positive definite at all critical points. Due to Lemma 6.3, at a critical point we have Consider the orthogonal to u support hyperplanes a 1 , b 1 and a 2 , b 2 of ϕ y (K 1 ) and ϕ y (K 2 ). They are images under ϕ y of support hyperplanes of K 1 and K 2 that are either parallel or share a (d − 2)-dimensional affine subspace. Since the cross-ratio is projectively invariant, (3) holds for a 1 , b 1 and a 2 , b 2 . By Lemma 6.5 we have which, by Lemma 6.4, implies Thus, at every critical point D 2 u,u F(y) > 0 for all u = 0. By Lemma 6.6, this implies that the critical point is unique.
Let us show that the bound (3) is sharp. Take as K 1 a double cone of height 1 over a (d −1)dimensional subset of R d−1 with centroid at the origin, and as K 2 a cylinder of height > κ d over a similar, but smaller, set. For example, K 1 may be the standard cross-polytope, and K 2 a rectangular parallelepiped. Then the centroids of K 1 and K 2 coincide, so that e 0 is a critical point of the function F. The quadratic form D 2 F(e 0 ) takes a negative value in the direction of the axis of K 1 and K 2 . Therefore e 0 is not the minimum point of F. Thus a minimum point provides a non-affine projective transformation that matches the centroids of K 1 and K 2 .

If one of the bodies is a ball
where S d−1 t is the (d − 1)-dimensional sphere of radius t, and ω d−1 is the volume of the unit (d − 1)-sphere. On the other hand, vol(B r ) = 1 d ω d−1 r d , which leads to the formula of the lemma.

Proof of Theorem 7
Similar to the proof of Theorem 6, it suffices to show that for all y.
The image of a ball under an admissible projective transformation is an ellipsoid. It is easily seen that the normalized moment of inertia I u / vol of an ellipsoid equals to the normalized moment of a ball with diameter equal to the width of the ellipsoid in direction u: Because of Lemmas 6.4 and 6.7 we have To ensure the latter inequality for all u, it suffices to require for all quadruples of parallel tangent hyperplanes to ϕ y (B d ) and ϕ y (K ). This, in turn, is implied by the same inequality for concurrent tangent hyperplanes to B d and K .

Proof of Theorem 8
Without loss of generality, assume 0 ∈ int K (this may be achieved by a projective transformation that fixes B, and the Theorem is of projective nature). By Sect. 2.3, K • y = ϕ y (K • ) + y, so that γ (K • y ) = γ (B • y ) ⇔ γ (ϕ y (K • )) = γ (ϕ y (B)) Use the same method as in the proofs of Theorems 6 and 7. The assumption of the theorem implies width(K • ) < log for all y, and in particular for critical points of the function F. Due to Lemmas 6.4 and 6.7, this implies that at the critical points F is strictly convex. Thus by Lemma 6.6 the critical point is unique.

Other dimensions
We restricted our attention to the cases dim K i ∈ {0, d} because it implies the affine covariance of the centroid (1), which makes it possible to formulate the uniqueness problem (projective transformations modulo affine ones). For dim K = k / ∈ {0, d} the centroid is affinely covariant under some additional restrictions, for example if K is centrally symmetric or if the affine span of K has dimension k.

Problem 1 What is the most general class of k-dimensional subsets of R d with affinely covariant centroids?
Note that the affine covariance of the centroid of K doesn't yet make the uniqueness question well-posed: in principle, one needs the affine covariance for all projective images of K . Problem 2 Study the existence and uniqueness questions when dim K i / ∈ {0, d} for at least one i. Do the solutions correspond to the critical points of some function F?

Point versus a body
By Theorem 2, every point inside a cusp-free set becomes the centroid after some projective transformation. Example 4.8 gives a point inside a set lying close to a sharp cusp so that no projective transformation can make it the centroid.

Problem 3
Weaken the cusp-freeness condition so that to preserve the existence of a projective transformation for any point in the interior. Is the following condition necessary and sufficient: the image of K under any projective transformation that sends some support hyperplane of K to infinity has infinite volume?

Problem 4 If K has sharp cusps, describe the set of all points that can become centroids. Is it an intersection of K with a convex body? Is it related to the Dupin's floating body?
We conjecture the following solution of the latter problem: for every support hyperplane of K take a projective transformation ϕ that sends to infinity. If ϕ (K ) has a finite volume, then cut off a half of the volume of ϕ (K ) by a hyperplane parallel to (remove the infinite part). The complement in K to the preimages of all parts removed in this way is the set of points that can become the centroid.

Several points versus a body
In Examples 5.3 and 5.2 we saw that the projective transformation matching the centroids of a finite set and of a "bigger" d-dimensional set does not always exist, and if, then may be not unique. By contrast, if the d-dimensional set is a ball, then we have unconditional existence and uniqueness.

Problem 5 Does a projective transformation always exist, if through every point on the boundary of conv K goes exactly one support hyperplane?
Problem 6 Is the projective transformation unique, if K is "round enough" in some sense?

If no body is contained in the other
Our method to prove the existence was to show that the function F tends to +∞ near the boundary of its domain. This was ensured by the assumption K 2 ⊂ int conv K 1 . It is possible to prove the existence of a critical point of F under less restrictive assumptions, for example, if d = 2 and the gradient of F "turns" as we go along the boundary of domain of F. This means that a projective transformation exists if K 2 "sticks out" of K 1 in at least two places. One could formalize and generalize this argument by using the degree of the map from the boundary of the domain of F to the (d − 1)-dimensional sphere.

Problem 7
Give a sufficient condition for the existence of a projective transformation in the case when neither K 1 ⊂ conv K 2 nor K 2 ⊂ conv K 1 .
Funding Open access funding provided by TU Wien (TUW).
Data availibility All data generated or analysed during this study are included in this published article.
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