On a formula for all sets of constant width in 3d

In the recent paper"On a formula for sets of constant width in 2D", Comm. Pure Appl. Anal. 18 (2019), 2117-2131, we gave a constructive formula for all 2d sets of constant width. Based on this result we derive here a formula for the parametrization of the boundary of bodies of constant width in 3 dimensions, depending on one function defined on S^2. Each such function gives a minimal value r_0 and for all r \ge r_0 one finds a body of constant width 2r. Moreover, we show that all bodies of constant width in 3d have such a parametrization. The last result needs a tool that we describe as 'shadow domain' and that is explained in an appendix. The construction is explicit and and offers a parametrization different from the one given by T. Bayen, T. Lachand-Robert and \'E. Oudet,"Analytic parametrization of three-dimensional bodies of constant width"in Arch. Ration. Mech. Anal., 186 (2007), 225-249.


Introduction
The interest in the subject started with Leonhard Euler, who around 1774 considered 2d curves of constant width, which he called 'curva orbiformis'.He not only studied such sets for 2 dimensions but also gave a formula describing such curves.See §10 of [5].In 3 dimensions a ball is obviously the classical example of a body of constant width but the famous Meissner bodies also have this property.See [17,18] or [14].Quite simple examples can also be constructed by taking a reflection symmetric 2d set of constant width and rotating it around its line of symmetry.
Famous mathematicians such as Minkowski [20] and Hilbert [10] were intrigued by the subject.The first interest of most scholars focused on deriving properties of such domains.A wonderful survey on sets of constant width (up to 1983) was provided by Chakerian and Groemer in [3], and a more recent updated and thorough treatment can be found in the book by Martini, Montejano and Oliveros [16].Let us recall that the 3d question, motivated by Blaschke's 2d result [2], as to which body of constant fixed width has the smallest volume or, equivalently, the smallest surface area, is still open.We will not solve that problem, but will give an alternative formula for constructing bodies of constant width that might help.
Let us recall some known facts about sets of constant width.Sets of constant width G in R n are strictly convex and hence any tangential plane touches G in at most one point.Moreover, for every boundary point X of G and r > d G there exists even a ball B r of radius r such that G ⊂ B r and ∂G ∩ ∂B r = X.So G slides freely in B r , according to the definition of [11, page 244].
Although the Gauss-map ∂Ω → S n−1 (the outward normal on smooth parts of ∂G) will not be uniquely defined on edges or corners, the 'inverse' γ G : S n−1 → ∂G is well-defined for a strictly convex G and parametrizes ∂G.See [15].In [16,Theorem 11.1.1]one finds, when G is a set of constant width and γ G ∈ C 1 S n−1 , that γ G (ω) = P G (ω) ω + ∇ P G (ω) for all ω ∈ S n−1 , ( where P G (ω) := max {⟨ω, x⟩ ; x ∈ G} is the support function and ∇ P G its gradient along S n−1 , i.e. ⟨∇ P G (ω), v⟩ = (dP G (ω)) (v) for all ω ∈ S n−1 and v ∈ T ω S n−1 .Howard in [11,Corollary 2.6] states that for a body of constant width the support function P G is of class C 1,1 and hence γ G is Lipschitz continuous.A direct proof of this Lipschitz-continuity also follows from Lemma 11 below.Necessary for a set G of constant width d G is that Therefore P G (ω) + P G (−ω) = d G and ∇ P G (ω) = ∇ P G (−ω) for all ω ∈ S n−1 .
In [1] a parametrization of sets of constant width is given by using the so-called median surface, which is parametrized by Writing x • Y := i x i Y i , which may coincide with but will not be restricted just to the inner product ⟨•, •⟩, the convexity of G leads to while (1.2) implies M G (ω) = M G (−ω) for all ω ∈ S n−1 . (1.5) The reverse question would be: can one give criteria on a continuous function γ : S n−1 → R n such that γ S n−1 parametrizes the boundary of a set of constant width?An answer is given by Theorem 2 of [1], where it is stated that for any continuous map M : S n−1 → R n and α > 0, which satisfy the set is of constant width d G := α and M G (ω) := M (ω).The γ G and M G are as in (1.3).One finds by continuity of M that and even that the identity holds in (1.8).
Continuity or even differentiability of M by itself is not enough for an α to exist for which (1.6) holds.The second condition in (1.6) implies the convexity of G from (1.7) and as such it gives a monotonicity for directional derivatives, hence a necessary one-sided estimate for second derivatives of M , whenever these exist.In two dimensions, see [13], a few simple conditions on a function in L ∞ (0, π) are necessary and sufficient in order to have a curve of constant width.The construction in 2 dimensions is also helpful in 3 dimensions.It will allow us to give a more explicit formula for all bodies of constant width, which is what we want to show here.

Two dimensions
In the last century Hammer and Sobczyk described a construction for 2 dimensions in [7,8,9], based on a characterization of what they called 'outwardly simple line families'.More recently a direct concise formula was given in [13] to describe all those sets in two dimensions starting from any L ∞ (0, π)function satisfying 2 equations, namely the ones in (2.3).Let us recall the 2d formula from [13]: − sin s cos s ds. (2.4) Then x describes the boundary of a set of constant width 2r.
For a simple statement in Theorem 1 the function a ∈ L ∞ (0, π) is extended to R and such that (2.1) and (2.2) are satisfied.The formula in (2.4) shows that x ∈ C 0,1 (R), which is optimal for r = ∥a∥ ∞ .For r > ∥a∥ ∞ when considering the set x([0, 2π]) as a curve one finds that x([0, 2π]) ∈ C 1,1 .The formula in (2.4) describes the boundary of all 2d domains of constant width: a closed convex set of constant width 2r, then there exists x 0 and a as in Theorem 1, such that ∂G = x ([0, 2π]) with x as in (2.4).
The geometric interpretation of the formula in (2.4) is that x(φ) and x(φ+π) describe the ends of a rotating stick of length 2r with the varying point of rotation lying on the stick by (2.1) and determined by a(φ).For these ends to coincide for φ ∈ [0, π] with those for φ ∈ [π, 2π] one needs condition (2.2).The two equalities in condition (2.3) make it a closed curve.

A formula in three dimensions
There have been previous attempts to provide an explicit construction of all 3d bodies of constant width.In [15] Lachand-Robert and Oudet present a geometric construction that generates 3d bodies of constant width from 2d sets of constant width.This construction, however, does not capture all 3d bodies of constant width because a counterexample is provided in the paper [4] by Danzer, who constructs a body of constant width d, none of whose planar cross-sections have constant width d in two dimensions.In [21] Montejano and Roldan-Pensado generalize the construction of Meissner bodies to generate so-called Meissner polyhedra.This construction does not generate all 3d bodies either, because the rotated Reuleaux triangle is a counterexample.As already mentioned Bayen, Lachand-Robert and Oudet give a description of (all) n-dimensional sets of constant width in [1, Theorem 2], but the function M has to satisfy a condition at each point of S n−1 .We provide an alternative construction, based on the method from [13], which gives a simpler condition although more involved than some integral conditions and an L ∞ bound.Indeed, some simple conditions as in 2d do not seem possible, but our conditions in 3d will come close.
Our approach uses spherical coordinates in R 3 .Indeed, for each fixed angle θ we apply the 2dapproach to get a curve parametrized by φ of constant width 2r.So as a first step the function a from Theorem 1 is now depending on θ φ → a(φ, θ) for each θ, and is used to define a curve φ → x(φ; θ), with x as in (2.4) and θ as a parameter, in the θ-dependent plane This first step however does not yet generate a body of constant width.Whenever ∂ 2 θ a ∞ is bounded and when r is large enough, the second step is to apply a unique shift in the perpendicular (− sin θ, cos θ, 0) ⊤ -direction for the collection of these rotating 2d-curves.For the magnitude of the shift we will use h(φ, θ).The combined result of these two steps will yield a 3d-body of constant width.Moreover we will show, that not only the result is a body of constant width but also that each such body can be written this way.
Aside from our results from [13] for two dimensions we will use a result by Hadwiger in [6], which can be roughly described as follows: convex bodies in R n are uniquely determined by the projections in R n−1 perpendicular to one fixed direction.The result holds for n ≥ 4 and, whenever the one fixed direction is regular, also for n = 3.This last addendum is due to [12].Regular means here, that the planes perpendicular to that fixed direction which touch the convex domain, do that in precisely one point.Since sets of constant width are necessarily strictly convex, this is obviously the case for those sets and any choice of the fixed direction.
Let us define for ω ∈ S 2 the orthogonal projection P ω on the plane E ω := x ∈ R 3 ; ⟨x, ω⟩ = 0 .To exploit the result of Hadwiger we will use for a fixed u ∈ S 2 all projections in the directions ω ∈ S 2 with ⟨ω, u⟩ = 0. See Fig. 1.For those ω we have For later use we need to identify the projections on E ω with coordinates in R 2 through Figure 1: The plane E ω for one ω and 'all' planes E ω with ω such that ⟨ω, u⟩ = 0.Those E ω contain u as a common direction.
We may now explain the result by Hadwiger in [6] in more detail.He proved that for two convex bodies G 1 and G 2 in R 3 the following holds.
Here A ≃ B means that A equals B after a translation.In other words, there is a fixed v ∈ R 3 such that A = v + B. Groemer showed in [12] that one could drop the condition P u G 1 ≃ P u G 2 , whenever u is a regular direction for G 1 .Here regular means that max {⟨u, x⟩ ; x ∈ G 1 } is attained for a unique x ∈ G 1 .Since domains G of constant width are precisely those domains for which This means that all those P ω G should be two-dimensional convex sets of constant width ρ.So by taking u = (1, 0, 0) we find that the boundary of P ω G is described by (2.4) with some a depending on ω.This leads us to the result in Theorem 5 This is the standard parametrization with φ the angle between ω and the positive z-axis and θ the counterclockwise angle of the projection on the xy-plane with the x-axis, viewed from the positive z-axis.Obviously this parametrization is not unique as we may restrict (φ, θ) to some subset of R 2 .
We may define a convenient φ, θ-dependent orthonormal basis, first for sin φ ̸ = 0, with the expression in the middle showing the obvious extension when sin φ = 0.
Any function (φ, θ) → v(φ, θ) : R 2 → R that is used to define a quantity on S 2 necessarily has to possess the obvious periodicity properties as well as some compatibility conditions.The relations, which the function a from (3.1) has to satisfy, are more subtle.For φ ̸ ∈ {0, π} the value r − a(φ, θ) coincides with the inverse curvature in the φ-direction.There is however a peculiarity at the north-and southpole, where the curvature in any(!) direction is given by (r − a(0, θ)) −1 , respectively (r − a(π, θ)) −1 , through varying θ.This leads to the following definition with a distinction between pure periodicity and what we call compatibility, both derived from U (φ, θ) = U ( φ, θ): Definition 4 For a function f : R 2 → R we say that: • f satisfies the periodicity conditions for S 2 , if • f satisfies the compatibility conditions for the poles of S 2 , if Suppose that B(R 2 ) is some function space.We write: ) and satisfies (3.6) and (3.7); ) and satisfies (3.6), (3.7) and (3.8).
With obvious changes we use the similar notations for a vector-valued F : R 2 → R 3 whenever f = F i satisfies the required properties for all i ∈ {1, 2, 3}.
One usually restricts R 2 to [0, π] × [0, 2π] to have a unique parametrization at least for the interior points and with some compatibility assumptions at its boundary, but here it will be more convenient to take As in the 2d-case the function a from (3.1) that we use is such that at opposite points of S 2 the value is opposite: Let V and W be as in (3.5) and suppose that h : (0, π) × R → R is defined by: 1. Then the definition in (3.12) can be continuously extended to R 2 .The extended h is such that and satisfies

.15)
2. There exists such that for all r ≥ r 0 (a) and X 0 ∈ R 3 , the surface X(S), defined by describes the boundary of a body of constant width 2r.
3.Moreover, with a(•, •) as above, the function h in (3.12) is the unique possibility in order that X in (3.17) describes the boundary of a body of constant width 2r.
Remark 5.1 Although a ∈ C 2 p (R 2 ) will imply that (3.15) holds, one finds at most X ∈ C 0,1 p,c (R 2 ).Hence the induced parametrization S 2 → ∂G is not necessarily a diffeomorphism.It will only be the a diffeomorphism for r > r 0 (a) and in general not for r = r 0 (a).By taking r > r 0 (a) one obtains a C 1,1 -surface with a distance ε = r − r 0 (a) from the body of constant width for r = −r 0 (a) where Lipschitz is optimal.The surface for r > r 0 (a) will also be a boundary for a body of constant width.Our construction will be illustrated by an example in Section 4.There the example has a ∈ C 1,1 p (R 2 ) and is such that r = 1.25348 ≈ r 0 (a), and for r = r 0 (a) the surface X will not be a diffeomorphism.The value of r 0 (a) can be computed numerically by finding the smallest r ≥ ∥a∥ ∞ such that T (r 0 ) ≥ 0 in (5.12) and D(r 0 , φ, θ) ≥ 0 in (5.13) holds for all φ, θ.Notice both are parabola in r with minima before ∥a∥ ∞ .
We have assumed that a ∈ C 2 p (R 2 ), which is sufficient for describing a 3d set of constant width for r large, but certainly more than necessary for h and X to be well-defined.Necessary for h to be well-defined will be L ∞ bounds for a, a θ and a θθ .For the 2d case a necessary and sufficient restriction appears, namely r ≥ r 0 (a) := ∥a∥ ∞ .In 3d this condition is still necessary but not sufficient.To have a differentiable parametrization in 3d a bound appears that contains ∂ θ h.We are however not able to quantify such a bound more precisely like in 2d.
We first prove some results for h that we gather in the next lemma.In fact, Lemma 6 contains the first item of Theorem 5.

.20)
Proof.A priori h is defined for φ ∈ (0, π) with the periodicity in the θ-direction being a consequence of the assumption that a satisfies (3.6) and (3.7).To consider the extension in the φ-direction first let us focus on the enumerator for h in formula (3.12).The enumerator is C 1 , since a is C 2 .By (3.11) we find that Moreover, for φ ∈ 0, 1 2 π we use 1 ≤ 1 + cos φ and find Hence we find that h can be continuously extended by 0 for φ ∈ {0, π} and Taking the formula in (3.12) for φ + π ∈ (π, 2π), we find by (3.11), a substitution and (3.10) and we find that the definition of h is well-defined on (π, 2π) × R and at least there (3.13) holds.Then h can be extended continuously by 0 and (3.18) holds for φ ∈ {0, π, 2π}.This allows us to use the definition in (3.12) for h for all φ with sin φ ̸ = 0 and to set h = 0 whenever sin φ = 0.Moreover, since a θ ∈ C 1 p (R 2 ) we find that h and also h W satisfies (3.6).One also finds that (3.14) holds true.For (3.7) note that and with we indeed find (3.7) for hW .Moreover, ) and hence (3.15) holds.Since the θ-dependence only comes through a the estimate in (3.20) is proven similarly as for (3.18).For (3.19) we use a straightforward computation from (3.12) and using (3.11) to find The estimate in (3.20) follows as the one in (3.18), which concludes the proof of Lemma 6.
Proofs of Theorem 5 and of the converse result in the next theorem are given in Section 5.

An example
The formulas are rather technical and in order to illustrate that (3.17) does deliver a body of constant width, we give an actual construction in a case that is computable.The example shows a body of constant width connecting two triangular 2d-domains of constant width based on the 2d-formula.In addition to x 0 = (0, 0) and r = 1 we use in Fig. 2: • for the figure on the left: a(s) = a 1 (s) := − cos(3s); • for the figure in the middle: a(s) = a 2 (s) := sin(3s).
One directly checks that conditions (2.2) and (2.3) are satisfied for a 1 and a 2 .
The object on the right of Fig. 2 combines these two curves in a 3d-setting in orthogonal planes with the red line as common intersection.In order to find a smooth perturbation from the horizontal to the vertical curve by curves whose projections will be 2d-curves of constant width 1, we use the following: a (φ, θ) := (cos θ) 2 a 1 (φ) + |sin θ| sin θ a 2 (φ).
Figure 2: The 2d sets with a 1 and a 2 for x(0) on top, and the combination in 3d by perpendicular planes and joining the red axes.
One has r 0 (a) ≥ max (r 0 (a 1 ), r 0 (a 2 )) = 1.Concerning the value of r 0 (a) for a in (4.1) a numerical estimate for the expression in (5.13) to be positive shows r 0 (a) ≈ 1.25348, which lies inside the interval given in (3.16).The a in (4.1) is used to produce the sketch on the left in Fig. 3 using the formula in (3.17) without the h-term.Each intersection with a plane containing the vertical (red) line {λ(0, 0, 1); λ ∈ R} will produce a 2d set of constant width.After the modification with the additional h-term in (3.17) does one indeed find a 3d set of constant width, which is found on the right of Fig. 3.
Although each body of constant width can be constructed through the formula in 3.17 it is relatively easy if one connect two curves of constant width as 4.1.More examples can be found in Fig. 9.

Proofs of the two theorems
For the standard inner product of u, v ∈ R n we use ⟨u, v⟩.The notation u • v is used for componentwise multiplication, which includes but can be more general than the inner product.Let us start by introducing three vectors for a more concise notation: These three directions constitute a θ-dependent orthonormal basis in R 3 that turns out to be convenient for our parametrization.The following identities hold true: Also note that our initial basis (3.5) can be expressed in term of (5.1): (5.4) The dot product • in (5.3), (5.4) is a more convenient notation in the following proofs.
Proof of Theorem 5. We will have to show that X in (3.17) is a regular parametrization and secondly, that the resulting surface will yield a body of constant width.For both aspects we need to consider ∂ φ X (φ, θ) and ∂ θ X (φ, θ).
▶ Computation of ∂ φ X (φ, θ) and ∂ θ X (φ, θ).We will check first that X in (3.17) is a regular parametrization of the boundary ∂G of a body of constant width for r large enough, that is is C 1 , one-to-one and onto, and even a diffeomorphism.With the notation from (5.1) we can rewrite (3.17) as One computes that and that ▶ Invariant normal direction.The next step is to show that the outward normal direction at X (φ, θ) satisfies: ν X(ω) = ±ω for all ω ∈ S 2 (5.9) where ω = U (φ, θ) as in (3.4).Indeed, we will first show that ω is perpendicular to ∂ φ X (φ, θ) and ∂ θ X (φ, θ).Taking h as in (3.12) is in fact the only possible choice such that holds.Indeed with this h we may rewrite (5.8), at least when sin φ ̸ = 0, to and using ω = cos φ Ξ + sin φ Θ one directly finds (5.10).As in the proof of Lemma 6 the factor in front of Ξ in (5.11) can be continuously extended by 0 when sin φ = 0. From (5.7) and again with If ∂ θ X or ∂ φ X is trivial for some (r, θ, φ) one may consider (r + ε, θ, φ) and find from (5.7) and (5.11) that for ε > 0 the corresponding expressions will be nontrivial and (5.10) will hold for that (φ, θ).
After the homotopy to the sphere furtheron, one may conclude that ω is the outside normal for (r + ε, θ, φ) with ε large and, by continuity, is an outside normal for X(R 2 ) at (θ, φ) when r > r 0 (a) with r 0 (a) to be defined in (5.15).▶ Well defined parametrization.For a ∈ C 2 p (R 2 ) Lemma 6 implies that h is well-defined and hW lies in C 1 p,c (R 2 ).So with (3.10) and (3.11) also the expression in (5.6) lies in C 1 p,c (R 2 ).In order to have a regular parametrization it is sufficient that: 1  2 π is nontrivial for φ ∈ {0, π}.Let us start with the second case for φ = 0, with φ = π similarly: (5.12) Note that T ′ (r) ≥ 0 for r ≥ ∥a∥ ∞ .
For φ ̸ ∈ πZ, using ( 5) and (5.10), which state that ω is perpendicular to ∂ φ X and ∂ θ X, a simple way of checking that Using the orthonormal basis {Θ, Ψ, Ξ} we obtain from (5.7) and (5.11):For φ ∈ (π, 2π) one obtains similar estimates for |D(r, φ, θ)| = −D(r, φ, θ).The expression in (5.13) can now be estimated.Using (3.19) and (3.20) from Lemma 6 we get for φ ̸ ∈ πZ: Moreover, whenever r ≥ ∥a∥ ∞ (5.13) also shows that ∂ r (D(r, φ, θ)/ sin φ) ≥ 0. Since T and |D| for φ ̸ ∈ πZ are increasing with respect to r for r ≥ ∥a∥ ∞ , there exists a minimal such that T and |D| for φ ̸ ∈ πZ are positive and hence that the parametrization is well-defined for all r > r 0 (a).For r = r 0 (a) the parametrization is no longer necessarily of class C 1 or one-to-one.However, since for all r > r 0 (a) one will find a body of constant width and all functions involved are continuous, also the limit by taking r ↓ r 0 (a) ̸ = 0 will give a body of constant width.
(5.16)So R 3 \ Xe r, S 2 has precisely two connected components.We call A the bounded one.
▶ Convexity of A. Since the extreme value of X(S 2 ) in the direction ω has normal ω, and since by (5.16) ν X(ω) = ω, that extreme point is indeed X (ω).So for each ω it holds that X(S 2 ), except for X (ω) itself, is on one side of that tangent plane.Hence Ā lies on one side of all the tangent planes for ∂A = X(S 2 ), which implies that Ā is convex.See also the proof of Hadamard's Theorem [19, page 194].
▶ Body of constant width.According to the results proved above it is sufficient to show that For the parametrization X with U as in (3.4) this coincides with Indeed, using (5.6) we find with (3.10), (3.11) and (3.13) that as desired.
Proof of Theorem 7, the derivation of formula (3.17) for some h.Suppose that G is a body of constant width d G .Define X 0 ∈ R 3 as the point on ∂G with the largest x 3 -coordinate.Since a translation that maps X 0 to a fixed point does not meddle with our arguments, we may assume (5.17) Taking u = (0, 0, 1) T and ω = (0, − sin θ, cos θ) T the result of Hadwiger, extended by the remark of Groemer that bodies of constant width have only regular boundary points, states that is is sufficient that the projections P ω G of G, on each of the planes spanned by {Θ(θ), Ξ} with θ ∈ [0, π], are curves of constant width d PωG = 2r.Thus by Theorem 2 all those sets can be described by (2.4) with for each θ some function a(•, θ) depending on θ as a parameter.The value of r is the same for all projections and does not depend on θ.In other words, a fixed r exists and for each θ a mapping φ → a(φ, θ) ∈ L ∞ (0, 2π) such that for the corresponding x as in Theorem 1 we have, with P as in (3.3), with some x 0 (θ) = x (0, θ) ∈ R 2 in accordance with Theorem 1 and Moreover, the mapping φ → a(φ, θ) satisfies (2.2) and (2.3).Hence (3.10), (3.11) and r 0 (a) ≥ ∥a∥ L ∞ (S) are necessary conditions.Since for each ω ∈ S 2 the set G lies in the cylinder perpendicular to its projection, in other words, we have there is (φ, θ) ∈ S with U (φ, θ) • ω = 0 and a value h (φ, θ) ∈ R such that with ω = Ψ(θ).If X * * ∈ ∂G is such that ∥X * − X * * ∥ = 2r, with 2r being the width, also ∥P ω (X * ) − P ω (X * * )∥ = 2r and hence with the same contribution h(φ, θ)ω, which implies that h (φ, θ) = h (φ + π, θ) for all (φ, θ) ∈ S. (5.18) Indeed Here (5.18) follows from the fact that the line through the points of farthest distance is perpendicular to the plane spanned by {Θ(θ), Ξ}.Since for φ ∈ {0, π, 2π} the X * in (5.19) does not depend on θ, one finds for all θ ∈ [0, π], that The first factor on the right in (5.19) inherits the conditions of the two-dimensional formula and so for each θ ∈ [0, π] one finds φ → x(φ, θ) as in (2.4).The formula in (5.19) describes through X : S → R 3 all points of ∂G by (5.20) We will also define . (5.21) Note that U (φ, θ) = cos φ Ξ + sin φ Θ(θ) lies in the plane of X oh (S) and is perpendicular to the cylinder X oh ([0, 2π], θ) + R Ψ(θ).Since G lies inside this cylinder and X(φ, θ) is a point of ∂G on this cylinder, the vector U (φ, θ) is an outwards normal to X(S) at X(φ, θ).In other words, writing X as in (5.5) it follows that ω → X(ω) : is the 'inverse' of the Gauss map for ∂G and hence Lipschitz-continuous on S 2 by Lemma 11.Note that Lemma 10 shows that Lipschitz-continuity of X on S 2 implies Lipschitz-continuity of X on S (but not vice versa!).So we may state, allowing the notation C 0,1 p (S) as a restriction of C 0,1 p (R 2 ), that for each coordinate in (5.20) the Lipschitz-continuity holds for: Combining the second function above multiplied with sin θ and the third multiplied with cos θ we find that h is Lipschitz-continuous on S, which is not sufficient for Lipschitz-continuity on S 2 .The function (φ, θ) → (sin θ, cos θ) transferred to ω turns into a function that is not even continuous on S 2 .We need another proof that is Lipschitz-continuous on S 2 and for that we will use the next two lemmata.We continue this proof on page 16.
Note that the sketch on the left of Fig. 3 shows a domain with boundary X oh (S).
For the proof, that ω → Xoh (ω) from (5.21) is Lipschitz-continuous, we will use the following: if X(S 2 ) describes the boundary of G, then Xoh (S 2 ) gives the boundary of the 3d-shadow domain Sh Ξ (G), when rotating around the central axis Ξ.The definition of 3d-shadow domain is found in Appendix C. The function X oh does not only parametrize the boundary of that 3d-shadow domain, but since we have that each shadow in the direction of Ψ(θ) has the contour parametrized by φ → X oh (φ, θ).
We start with an a-priori estimate for the position of ∂G in relation with the axes through the highest and lowest point of G.
Lemma 8 Let X be as in (5.20) with X 0 as in (5.17).Then for each (φ, θ) ∈ S one finds: (5.23) Proof.In any horizontal direction the boundary ∂G lies between the extreme cases of two-dimensional curves of constant width.These extreme cases are the Reuleaux triangle pointing left and the one pointing right.See Fig. 4. Rotating the left image around the vertical axis gives the area on the right, where ∂G is located.
Here P is as in (3.2), Ψ(θ), Ξ as in (5.1), U as in (3.4) and Sh Ξ (Ω) is the 3d-shadow as in Definition 14 and the rotation with respect to the axis Ξ.
Proof.We still assume (5.17).From our construction one finds that the function Xoh : S 2 → R 3 parametrizes the collection of boundaries of '2d-shadows' in the directions Ψ(θ) for θ ∈ [0, π] and gives a bounded two-dimensional manifold in R 3 .Each 2d-shadow P Ψ(θ) (G) for θ ∈ [0, π] is a twodimensional set of constant width in the plane spanned by Ξ and Θ(θ).The 3d-domain Ω bounded by these curves, that is is in general not a body of constant width and not even convex.But by rotating a body G of constant width around Ξ, we may use that each projection on the {Θ(θ), Ξ}-plane is a curve of constant width.
To show the weighted Lipschitz continuity estimate as in (5.31) for h, we use the expression and the auxiliary term Z(φ, θ) as in (5.24).We find In a similar way, we may show the estimate replacing sin φ by sin φ 0 and hence with Lemma 10 it follows that ω → h(ω) as in (5.22) is Lipschitz-continuous on S 2 .
Next we will derive the formula for h.Note that for Θ and Ψ as functions of θ: When there is no misunderstanding we skip the θ-dependence of Θ and Ψ and use only Θ = Θ(θ) and Ψ = Ψ(θ) .Thus one computes As X(φ, θ) describes the surface of a body of constant width 2r and we find that for all t ∈ R Note that and thus we necessarily have (5.32) Since U (φ, θ) = cos φ Ξ + sin φ Θ we find, using the Lipschitz-continuity of θ → X(φ, θ), that A On the distance in S 2 Let f : S 2 → R be some function.The standard definition for such a function f to be Lipschitzcontinuous, is, that there exists L > 0 such that Since the functions we use are defined in terms of (φ, θ) ∈ S instead of ω ∈ S 2 , with S from (3.9), we need to reformulate the Lipschitz-condition in (A.1) to a condition for with U from (3.4).In other words, we have to replace |ω − ω 0 | by an equivalent expression using (φ, θ) and (φ 0 , θ 0 ).The corresponding estimates follow next.

B The inverse Gauss map for bodies of constant width
As we mentioned in the introduction the Gauss map for bodies of constant width G is not necessarily uniquely defined on ∂G, but the inverse is.This 'inverse' is even Lipschitz and this can be found as a corollary in [11].In the next Lemma we will give a short direct proof.
Lemma 11 If X 1 and X 2 are two points on the surface of a body G ⊂ R 3 with constant width d G and ω 1 , ω 2 are outside normal directions at X 1 , X 2 , then Proof.Let ℓ : S 2 × S 2 → [0, π] be the distance function on the sphere S 2 , that is Since G is a body of constant width d G it holds for any X ∈ ∂G with outside normal ω that Here B r (M ) is the closed ball of radius r and center M .We call ω ∈ S 2 an outside normal at P ∈ ∂G if x • ω ≤ P • ω for all x ∈ G. Set Except for the circle C where the two spheres intersect, there is a unique outside normal direction on ∂L.There is a band on S 2 that contains the outside normal directions connected to the circle C. The circle in the middle of the band in S 2 we call S 0 .The shortest path from ω 1 to ω 2 on S 2 intersects S 0 at some ω m .There is a unique P ∈ C that connects with ω m .Set ω a to be the outside normal at P with respect to B 1 and ω b to be the outside normal at P with respect to B 2 .

C Shadow domains
In order to show that a body of constant width has some minimal regularity property, namely a kind of Lipschitz-continuity under rotation, we need a geometrical argument.Such an argument follows from 'observing the shadows' during rotation.We did not find such a tool in the literature and supply it here.
Definition 12 Suppose that Ω ⊂ R 2 is a bounded, simply connected domain with 0 ∈ Ω.We define R Ω : R → R + by R Ω (ψ) := sup x cos ψ + y sin ψ;  With the Ξ-axis being fixed the 3d-shadow is constructed as in the 2d-case for each Ξ-coordinate being constant.One obtains a 3d-domain by joining the 2d-shadows from rotating around that axis.We use Ξ as before but the 3d-shadow can be defined in any direction.

For a compact
set G ⊂ R n one defines its directional width in direction ω ∈ S n−1 := {x ∈ R n ; |x| = 1} by d G (ω) = max {⟨ω, x⟩ ; x ∈ G} − min {⟨ω, x⟩ ; x ∈ G} ,with ⟨•, •⟩ denoting the standard inner product.If G is convex and d G (ω) = d G is constant, then G is called a set of constant width.In 3 dimensions a set of constant width is also called a body of constant width.

Figure 3 :
Figure 3: On the left the intermediate construction still without the h.It consists of a rotating family of 2d-sets of constant width for each θ.This is not a body of constant width and not even convex.On the right is the corresponding body of constant width as the final result with the shift by h in the direction Ψ from (5.1).Both red curves originate from the curves from Fig.2.The surface on the right does not look smooth everywhere as indeed here r = 1.25348, the numerical approximation of r 0 (a).These red curves give two planar curves of constant width, since ∂ θ a(φ, θ) = 0 for all (φ, θ) ∈ R × πZ.

Figure 4 :
Figure 4: The axis in red with the shaded parts showing the possible areas for the projections in 2d.The shaded parts are the union of two Reuleaux triangles minus their intersection.So with the assumption r = 1 and |z| ≤ r = 1 we find for the picture on the left L ∈ [− √ 3, √ 3].If a body of constant width has (0, 0, ±1) on its boundaries, then the domain lies in the rotated shaded part, which is sketched on the right.

Figure 6 :
Figure 6: A sketch of S deformed to show a distance equivalent to the one in S 2 identifying points at the upper boundary as well as points on the lower boundary.For (A.2) see left and for see right.The variable φ ∈ [0, 2π] moves from left to right; θ |sin φ| with θ ∈ [− 1 2 π, 1 2 π] increases in the vertical direction.

Figure 8 : 3 
Figure 8: On the left a triangle, in the middle ψ → R Ω (ψ) as the maximum of the three functions related to the corners under counterclockwise rotation, and on the right the rotational shadow domain of the triangle − s) ds − sin φ h (φ, θ) + O t 2 ≤ 0. − s) ds − sin φ h (φ, θ) = O (t) .