Smoothness in the $l_p$ Minkowski problem for $p<1$

We discuss the smoothness and strict convexity of the solution of the $L_p$ Minkowski problem when $p<1$ and the given measure has a positive density function.

In this paper we are interested in this problem when p < 1 and ν is a measure with density with respect to the Hausdorff measure H n−1 on S n−1 , i.e. in the problem (1.1) dS K,p = f dH n−1 on S n−1 , where f is a non-negative function in S n−1 . According to Chou and Wang [15], if −n < p < 1 and f is bounded from above and below by positive constants, then (1.1) has a solution. More general existence results are provided by the recent works Chen, Li and Zhu [10] if p = 0, Chen, Li and Zhu [11] if 0 < p < 1, and Bianchi, Böröczky and Colesanti [3] if −n < p < 0 and 0 < p < 1.
We observe that h is a non-negative positively 1-homogeneous convex function in R n which solves the Monge-Ampère equation (1.2) h 1−p det(∇ 2 h + hI) = nf on S n−1 in the sense of measure if and only if h is the support function of a convex body K ∈ K n 0 which is the solution of (1.1) (see Section 2). Naturally, if h is C 2 , then (1.2) is a proper Monge-Ampère equation. The function h may vanish somewhere for certain functions f , and when this happen and p < 1 the equation (1.2) is singular at the zero set of h.
In this paper we study the smoothness and strict convexity of a solution K ∈ K n 0 of (1.1) assuming τ 2 > f > τ 1 for some constants τ 2 > τ 1 > 0. Concerning these aspects for p < 1, we summarize the known results in Theorem 1.1, and the new results in Theorem 1.2. We say that x ∈ ∂K is a smooth point if there is a unique tangent hyperplane to K at x and that K is smooth if each x ∈ ∂K is smooth (see Section 2 for all definitions). For z ∈ ∂K, the exterior normal cone at z is denoted by N (K, z), and for z ∈ int K, we set N (K, z) = {o}. Theorem 1.1 (i) and (ii) are essentially due to Caffarelli in [7] (see Theorem 3.6), and Theorem 1.1 (iii) is due to Chou, Wang [15].
If the function f in (1.1) is C α for α > 0, then Caffarelli [8] proves (iv). Theorem 1.1 (Caffarelli, Chou, Wang). If K ∈ K n 0 is a solution of (1.1) for n ≥ 2 and p < 1, and f is bounded from above and below by positive constants, then the following assertions hold: (i) The set X 0 of the points x ∈ ∂K with N (K, x) ⊂ N (K, o) is closed, any point of X = ∂K\X 0 is smooth and X contains no segment. (ii) If o ∈ ∂K is a smooth point, then K is smooth. (iii) If p ≤ 2 − n, then o ∈ int K, and hence K is smooth and strictly convex. (iv) If o ∈ int K and the function f in (1.1) is positive and C α , for some α > 0, then ∂K is C 2,α .
Concerning strict convexity Claim (iii) here is optimal because Example 4.2 shows that if 2 − n < p < 1, then it is possible that o belongs to the relative interior of an (n − 1)-dimensional face of a solution K of (1.1) where f is a positive continuous function. Therefore the only question left open is the smoothness of the solution if 2 − n < p < 1.
We note that if p < 1 and K is a solution of (1. 2) with f positive and o ∈ ∂K, then Therefore Theorem 1.1 (ii) yields that the solution K is smooth if n = 2. In general, we have the following partial results.
Theorem 1.2. If K ∈ K n 0 is a solution of (1.1) for n ≥ 2 and p < 1, and f is bounded from above and below by positive constants, then the following assertions hold: (i) If n = 2, n = 3 or n > 3 and p < 4 − n, then K is smooth. (ii) If H n−1 (X 0 ) = 0 for the X 0 in Theorem 1.1 (i), then K is smooth.
Our results differ in some cases from the ones in Chou and Wang [15], possibly because [15] considers the equation instead of (1.2). In the context of non-negative convex functions being a solution of this last equation is a priori more restrictive than being a solution of (1.2), even if obviously the two notions coincide where h is positive (see Section 2 for more on this point). Chou and Wang [15] proves, under our same assumptions on f , the strict convexity of the solution h of (1.4), and uses this to prove that the body K is smooth. In our opinion (1.2) is the right equation to consider and using it we obtain weaker results.
To give an example of the differences of the two equations, the support function h of the body K in Example 4.2 (where o belongs to the relative interior of an (n − 1)-dimensional face) is a solution of (1.2) but it is not a solution of (1.4).
According to Chou and Wang [15] (see also Lemma 3.1 below), the Monge Ampère equation (1.2) can be transferred to a Monge-Ampère equation for a convex function v on R n−1 where g is a given non-negative function. The proofs of Claims (i) and (ii) in Theorem 1.1 use as an essential tool a result proved by Caffarelli in [7] regarding smoothness and strict convexity of convex solutions of certain Monge-Ampére equation of type (1.5) (see Theorem 3.6). Proving that ∂K is C 1 is equivalent to prove that h K is strictly convex, and [7] is the key to prove this property in {y ∈ S n−1 : h(y) > 0}.
The proof of Claim (i) in Theorem 1.2 is based on the following result for the singular inequality v 1−p det ∇ 2 v ≥ g.
in the sense of measure, and S is r-dimensional, for r ≥ 1, then p ≥ −n + 1 + 2r.
The underlying idea behind the proof of this result is that on the one hand, the graph of v near S is close to be ruled, hence the total variation of the derivative is "small", and on the other hand, the total variation of the derivative is "large" because of the Monge-Ampere inequality.
The inequality p ≥ −n + 1 + 2r in this result is optimal, at least when r = 1. Indeed Example 3.2 shows that for any p > −n + 3 there exists a non-negative convex solution of (1. 6) in Ω which vanish on the intersection of Ω with a line. Proposition 1.3 yields actually somewhat more than Claim (i) in Theorem 1.2; namely, if r ≥ 2 is an integer, p < min{1, 2r − n} and K ∈ K n 0 is a solution of (1.1) with o ∈ ∂K, then dim N (K, o) < r. As a consequence, we have the following technical statements about K, where we also use Theorem 1.2 (ii) for Claim (ii). Corollary 1.4. If p < 1 and K ∈ K n 0 , n ≥ 4, is a solution of (1.1) with o ∈ ∂K, then (i) dim N (K, o) < n+1 2 ; (ii) if in addition n = 4, 5 and K is not smooth, then dim N (K, o) = 2 and dim F (K, u) = n − 1 for some u ∈ N (K, o).
We review the notation used in this paper in Section 2. Section 3 contains results and examples regarding Monge-Ampère equations in R n , namely Proposition 1.3, Example 3.2 and Proposition 3.4. This last result is the key to prove Theorem 1.2 (ii). In Section 4 we show, for the sake of completeness, how to prove Theorem 1.1 using ideas due to Caffarelli [7,8] and Chou and Wang [15]. Theorem 1.2 and Corollary 1.4 are proved in Section 5.

Notation and preliminaries
As usual, S n−1 denotes the unit sphere and o the origin in the Euclidean nspace R n . If x, y ∈ R n , then x, y is the scalar product of x and y, while x is the euclidean norm of x. By [x, y] we denote the segment with endpoint x and y.
We write H k for k-dimensional Hausdorff measure in R n . We denote by ∂E, intE, clE, and 1 E the boundary, interior, closure, and characteristic function of a set E in R n , respectively. The symbols affE and linE denote respectively the affine hull and the linear hull of E. The dimension dim E is the dimension of affE. With the symbol E | L we denote the orthogonal projection of E on the linear space L.
For notions and facts about Monge-Ampère equations, see the survey Trudinger and Wang [50]. Given a function v defined on a subset of R n , ∇v and ∇ 2 v denote its gradient and its Hessian, respectively. When v is a convex function defined in an open convex set Ω, the subgradient ∂v(x) of v at x ∈ Ω is defined as The associated Monge-Ampère measure is defined by For p < 1 and non-negative g on R n , we say that the non-negative convex function v satisfies the Monge-Ampère equation A convex body in R n is a compact convex set with nonempty interior. The treatise Gardner [19], Gruber [20], Schneider [46] are excellent general references for convex geometry. The function for u ∈ R n , is the support function of K. When it is clear the convex body to which we refer we will drop the subscript K from h K and write simply h. Any convex body K is uniquely determined by its support function.
If S is a convex set in R n , then a z ∈ S is an extremal point if z = αx 1 +(1−α)x 2 for x 1 , x 2 ∈ S and α ∈ (0, 1) imply x 1 = x 2 = z. We note that if S is compact convex, then it is the convex hull of its extremal points. Next let C be a convex cone; namely, is a closed convex cone such that the origin is an extremal point of C, then C is the convex hull of its extremal rays.
The normal cone of a convex body K at z ∈ K is defined as This definition can be written also as In particular, N (K, z) is a closed convex cone such that the origin is an extremal point, and is a ray, and K is a smooth convex body if each p ∈ ∂K is a smooth point. In the latter case, ∂K is C 1 , which is equivalent to saying that the restriction of h K to any hyperplane not containing o is strictly convex, by (2.3).
We say that K is strictly convex if ∂K contains no segment, or equivalently, h K is C 1 on R n \{o} (see (2.4)).
The face of K with outer normal u ∈ R n is defined as In particular, for any Borel ω ⊂ S n−1 , the surface area measure S K satisfies and hence S K is the analogue of the Monge-Ampère measure for the restriction of h K to S n−1 . Given a convex body K containing o and p < 1, let S K,p denote the L p area measure of K; namely, In particular, for a positive measurable f : S n−1 → R, h K is a solution of (1.2) in the sense of measure if and only if the following conditions (a) and (b) hold: (a) dim N (K, o) < n; or equivalently, Let us compare these two conditions to the conditions for h K , K ∈ K n 0 , being a solution of (1.4) for p < 1 and positive f . On the one hand, we have (2.6) and (2.7). However, since the exponent p − 1 < 0, we have to add the condition In particular, if K ∈ K n 0 is a solution of (1.4) for p < 1 and f is bounded from below and above by positive constants, then combining (2.8), Theorem 1.1 (i) and Theorem 1.2 (ii) shows that K is smooth, as it was verified by Chou and Wang [15].

Some results on Monge-Ampère equations in Euclidean space
Lemma 3.1 is the tool to transfer the Monge-Ampère equation (1.2) on S n−1 to a Euclidean Monge-Ampère equation on R n−1 . For e ∈ S n−1 , we consider the restriction of a solution h of (1.2) to the hyperplane tangent to S n−1 at e. Lemma 3.1. If e ∈ S n−1 , h is a convex positively 1-homogeneous non-negative function on R n that is a solution of (1.2) for p < 1 and positive f , and v(y) = h(y + e) holds for v : e ⊥ → R, then v satisfies where, for y ∈ e ⊥ , we have which is a possibly empty spherically convex compact set whose spherical dimension is at most n − 2, by (2.6). According to (2.7), the Monge-Ampère equation for h K can be written in the form which is induced by the radial projection from the tangent hyperplane e + e ⊥ to S n−1 . Since π(x), e = (1 + x 2 ) −1 2 , the Jacobian of π is . For x ∈ e ⊥ , (2.4) and writing h K in terms of an orthonormal basis of R n containing e, yield that v satisfies ∂v(x) = ∂h K (x + e)|e ⊥ = F (K, x + e)|e ⊥ = F (K, π(x))|e ⊥ .
Let S = π −1 ( S). For a Borel set ω ⊂ e ⊥ \S, we have where we used at the last step that ). In particular, v satisfies the Monge-Ampère type differential equation Proof of Proposition 1.3. Up to restricting Ω and changing coordinate system, we may assume, without loss of generality, that Ω = {(x 1 , x 2 ) ∈ R r × R n−r : x 1 < s 1 , x 2 < s 2 } and that S = {(x 1 , x 2 ) : x 2 = 0}, and v is continuous on cl Ω.

The function v is invariant with respect to rotations around the line containing S.
To compute det ∇ 2 v at an arbitrary point, it suffices to compute it at (x 1 , 0, . . . , 0, r), The function v is convex if β is sufficiently small. Indeed, the eigenvalues of ∇ 2 v are 1 r + f ′ (r) r g(x 1 ), with multiplicity n − 2, and those of the matrix The determinant of the latter matrix is which is positive if β > 0 is sufficiently small. Thus all eigenvalues of ∇ 2 v are positive.
We get which has the same order as r 2α−n as r → 0 + . Clearly v has order r, and v 1−p det ∇ 2 v has order r 2α−n+1−p , which is uniformly bounded from above and below for our choice of α.
The next statement is a slight revision of Lemmas 3.2 and 3.3 from Trudinger and Wang [50]. We remark that Lemma 3.2 in [50] proves (3.7) with sup Ω |v| instead of |v(o)|. The inequality (3.7) follows from that and the observation that if u is any convex function in Ω, which vanishes on ∂Ω, and tE ⊂ Ω ⊂ E then |u(o)| ≥ t/(t + 1) sup Ω |u|.  The proof of Claim (ii) in Theorem 1.2 is based on the following proposition. This proposition is related to a step in the proof of Theorem E (a) in [15], however the argument that we use to prove it is substantially different from that in [15].

Proposition 3.4.
Let v be a non-negative convex function defined on the closure of an open convex set Ω ⊂ R n , n ≥ 2, such that S = {x ∈ Ω : v(x) = 0} is non-empty and compact, and v is locally strictly convex on Ω\S. Let ψ : (0, ∞) → [0, ∞) be monotone decreasing and not identically zero; assume that τ 2 > τ 1 > 0 and v satisfy in the sense of measure on Ω\S. If dim S ≤ n − 1 and µ v (S) = 0 for the associated Monge-Ampère measure µ v , then S is a point.
Note that (3.8) means that for each Borel set ω ⊂ Ω \ S we have where µ v has been defined in (2.1).
Proof. We may assume that Ω is bounded. We suppose that dim(aff S) ≥ 1, and seek a contradiction. We may assume that o is the center of mass of S, let L = aff S = lin S and let e = (o, 1) ∈ R n × R. Let ε 0 > 0 be the minimum of v on ∂Ω and let us consider the convex body Since Ω is bounded and v is locally strictly convex on Ω \ S, no supporting hyperplane to M intersects both S and the top facet F (M, e). Therefore there exists a We write l ε to denote the linear function on R n whose graph is H ε , and define Since Ω is bounded, we have cl Ω ε ⊂ Ω by the choice of ε 1 . Let us prove that for each w ∈ L ⊥ ∩ R n we have (3.9) Ωε x, w dx = 0.
Indeed, for t ∈ R with |t| small, let By definition of l ε , F has a local minimum at t = 0. We have x, w dx and there exists c independent on t such that H n (A t ) < ct and sup At |l ε (x)−v(x)| < ct. Therefore we have dF dt (0) = Ωε x, w dx, which proves (3.9). Note that (3.9) implies that the center of mass of Ω ε is contained in L. Therefore (see [46,Lemma 2.3.3]) Ω ε contains the reflection of Ω ε with respect to this center of mass, scaled, with respect to the same center of mass, by a factor 1/n. We deduce from this that It follows from the definition of Ω ε that (3.11) S ⊂ Ω ε and lim where the last limit is in the sense of the Hausdorff distance. In particular, there exists ε 2 ∈ (0, ε 1 ) such that if ε ∈ (0, ε 2 ), then We observe that We claim that for any ε ∈ (0, ε 2 ), there exists an ellipsoid E ε centered at the origin such that (3.14) 1 According to Loewner's or John's theorems, there exists an ellipsoid E centered at the origin and z 1 ∈ Ω ε such that It follows from (3.10) that there exists z 2 ∈ Ω ε such that z 2 |L ⊥ = −1 n z 1 |L ⊥ . In particular, y 1 = 1 n+1 z 1 + n n+1 z 2 ∈ Ω ε satisfies that y 1 |L ⊥ = o, or in other words, y 1 ∈ L ∩ Ω ε . In addition, Let m = dim L ≤ n − 1. Since y 1 ∈ L ∩ Ω ε and (3.12) imply 1 2 y 1 ∈ S, and the origin is the centroid of S, we deduce that y 2 = −1 2m y 1 ∈ S. As 2m + 1 < 2n, we have 1 As Ω ε ⊂ 2 E follows from o ∈ z 1 + E, we may choose E ε = 2 E, proving (3.14). Let us apply Lemma 3.3 to Ω ε and to the function v − l ε . Let ν denote the Monge-Ampère measure µ (v−lε) restricted to Ω ε . If Ω 0 is an open set such that Ω ε ⊂ Ω 0 ⊂ cl Ω 0 ⊂ Ω then the set N v (Ω 0 ) is bounded and this implies ν(Ω ε ) = H n (N (v−lε) (Ω ε )) ≤ H n (N v (Ω 0 )) < ∞.
Let us prove that ν(tΩ ε ) ≥ bν(Ω ε ) if b = τ 1 t n /τ 2 . The function v is convex and attains its minimum at o, thus v(x) ≥ v(tx) for any x ∈ Ω ε . By this, the monotonicity of ψ, (3.8) and the assumptions on S, we deduce that Let c 0 and c 1 be the constants appearing in Lemma 3.3. It follows from (3.13) and Lemma 3.3 (ii) that if ε ∈ (0, ε 2 ), then On the other hand, let s = c n 1 /(2c 0 ) n . It follows from (3.11), dim S ≥ 1 and from the fact that the origin is the centroid of S that there exists ε ∈ (0, ε 1 ) small enough, such that S ⊂ L ∩ Ω ε ⊂ (1 + s)S. In particular, there exists z ε ∈ S such that (z ε + sE ε ) ∩ ∂Ω ε = ∅. It follows from Lemma 3.3 (i) that This contradicts (3.15), and in turn proves Proposition 3.4.
We will actually use the following consequence of Proposition 3.4.
If S = {x ∈ Ω : v(x) = 0} is non-empty, compact and µ v (S) = 0, and v is locally strictly convex on Ω\S, then S is a point.
Proof. All we have to check that dim S ≤ n − 1. It follows from the fact that the left hand side of the differential equation is zero on S, while the right hand side is positive.
The following result by Caffarelli (see Theorem 1 and Corollary 1 in [7]), handles the part of the boundary of a convex body K where the support function at some normal vector is positive.  We recall that (3.16) is equivalent to saying that for each Borel set ω ⊂ Ω we have λ 1 H n (ω) ≤ µ v (ω) ≤ λ 2 H n (ω), where µ v has been defined in (2.1).

Proof of Theorem 1.1
The next lemma provides a tool for the proof of Theorem 1.1 (iii). The same result is also proved in Chou and Wang [15]; we present a short argument for the sake of completeness.
Lemma 4.1. For n ≥ 2 and p ≤ 2 − n, if K ∈ K n 0 and there exists c > 0 such that S K,p (ω) ≥ c H n−1 (ω) for any Borel set ω ⊂ S n−1 , then o ∈ int K.
Proof. We suppose that o ∈ ∂K and seek a contradiction. We choose e ∈ N (K, o)∩ S n−1 such that {λe : λ ≥ 0} is an extremal ray of N (K, o). Let H + be a closed half space containing Re on the boundary such that N (K, o) ∩ intH + = ∅. Let B n be the unit ball centered at the origin o, and let It follows by the condition on S K,p that However, since h K is convex and h K (e) = 0, there exists c 0 > 0 such that We observe that the radial projection of V 0 onto the tangent hyperplane e + e ⊥ to S n−1 at e is e + V ′ 0 for If y ∈ V ′ 0 , then u = (e + y)/ e + y verifies u − e ≥ y /2. It follows that as p ≤ 2 − n. This contradicts (4.1), and hence verifies the lemma.
Proof of Theorem 1.1. Claim (i). For u 0 ∈ S n−1 \N (K, o), we choose a spherically convex open neighbourhood Ω 0 of u 0 on S n−1 such that for any u ∈ cl Ω 0 , we have u, u 0 > 0 and u ∈ N (K, o). Let Ω ⊂ u ⊥ 0 be defined in a way such that u 0 + Ω is the radial image of Ω 0 into u 0 + u ⊥ 0 , and let v be the function on Ω defined as in Lemma 3.1 with h = h K . Since h K is positive and continuous on cl Ω, we deduce from Lemma 3.1 that there exist λ 2 > λ 1 > 0 depending on K, u 0 and Ω 0 such that on Ω. First we claim that We suppose that dim N (K, z) ≥ 2, and seek a contradiction. Since N (K, z) is a closed convex cone such that o is an extremal point, the property N (K, z) ⊂ N (K, o) yields an e ∈ (N (K, z) ∩ S n−1 )\N (K, 0) generating an extremal ray of N (K, z). We apply the construction above for u 0 = e. The convexity of h K and (2.2) imply h K (x) ≥ z, x for x ∈ R n , with equality if and only if x ∈ N (K, z). We define S ⊂ Ω by S + e = N (K, z) ∩ (Ω + e) and hence o is an extremal point of S. It follows that the functionṽ defined byṽ(y) = v(y) − z, y + e is non-negative on Ω, satisfies (4.2), and S = {y ∈ Ω :ṽ(y) = 0}.
These properties contradict Caffarelli's Theorem 3.6 (i) as o is an extremal point of S, and in turn we conclude (4.3). Next we show that We apply again the construction above for u 0 . If u ∈ Ω 0 and z ∈ F (K, u) clearly K is smooth at z (i.e. N (K, z) is a ray) by (4.3). Therefore, by (2.3), v is strictly convex on Ω and Caffarelli's Theorem 3.6 (ii) yields that v is C 1 on Ω. In turn, we conclude (4.4). In addition, F (K, u) is a unique smooth point for u ∈ Ω 0 (see (2.4)), yielding that Ω * = ∪{F (K, u) : u ∈ Ω 0 } is an open subset of ∂K. Therefore Ω * ⊂ X, any point of Ω * is smooth (by (2.3)) and Ω * contains no segment (by (2.4)), completing the proof of Claim (i).
Claim (ii). We suppose that o ∈ ∂K is smooth, and there exists z ∈ ∂K such that K is not smooth at z. Claim (i) yields that z ∈ X 0 , and hence N (K, z) ⊂ N (K, o), which is a contradiction, verifying Claim (ii).
Claim (iii). This is a consequence of Lemma 4.1 and Claim (i). Claim (iv). This is a consequence of Lemma 3.1, Claim (i) and Caffarelli [8].
Example 4.2. If n ≥ 2 and p ∈ (−n + 2, 1), then there exists K ∈ K n 0 with smooth boundary such that o lies in the relative interior of a facet of ∂K and dS K,p = f dH n−1 for a strictly positive continuous f : S n−1 → R.
To prove that dS K,p = f dH n−1 for a positive continuous f : S n−1 → R, it suffices to prove that there is a neighborhood of the South pole where dS K,p /dH n−1 is continuous and bounded from above and below by positive constants. Let h be the support function of K and, for y ∈ R n−1 , let v(y) = h(y, −1) be the restriction of h to the hyperplane tangent to S n−1 at the South pole. It suffices to prove that in a neighborhood U of o, v satisfies the equation v 1−p det ∇ 2 v = G with a function G which is bounded from above and below by positive constants.
If y ∈ U \ {o} we have which coincides with X 0 by Theorem 1.1 (i), has (n − 1)-dimensional Hausdorff measure equal to zero and, by Theorem 1.2 (ii), K is smooth.