MINKOWSKI PROBLEM FOR POLYTOPES

Transcription

1 ON THE L MINKOWSKI PROBLEM FOR POLYTOPES DANIEL HUG, ERWIN LUTWAK, DEANE YANG, AND GAOYONG ZHANG Abstract. Two new aroaches are resented to establish the existence of olytoal solutions to the discrete-data L Minkowski roblem for all > 1. As observed by Schneider [21], the Brunn-Minkowski theory srings from joining the notion of ordinary volume in Euclidean d-sace, R d, with that of Minkowski combinations of convex bodies. One of the cornerstones of the Brunn-Minkowski theory is the classical Minkowski roblem. For olytoes the roblem asks for the necessary and sufficient conditions on a set of unit vectors u 1,..., u n S d 1 and a set of real numbers α 1,..., α n > 0 that guarantee the existence of a olytoe, P, in R d with n facets whose outer unit normals are u 1,..., u n and such that the facet whose outer unit normal is u i has area (i.e., (d 1)-dimensional volume) α i. This roblem was comletely solved by Minkowski himself (see Schneider [21] for reference): If the unit vectors do not lie on a closed hemishere of S d 1, then a solution exists if and only if α i u i = 0. i=0 In addition, the solution is unique u to a translation. In the middle of the last century, Firey (see Schneider [21] for references) extended the notion of a Minkowski combination of convex bodies and for each real > 1 defined what are now called Firey-Minkowski L combinations of convex bodies. A decade ago, in [11], Firey- Minkowski L combinations were combined with volume and the result was an embryonic L Brunn-Minkowski theory often called the Brunn-Minkowski-Firey theory. During the ast decade various elements of the L Brunn-Minkowski theory have attracted increased attention (see e.g. [3], [4], [5], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [22], [23], [24], [25], [26], [27]). A central roblem within the L Brunn-Minkowski theory is the L Minkowski roblem. A solution to the L Minkowski roblem when the data is even was given in [11]. This solution turned out to be a critical ingredient in the recently established L affine Sobolev inequality [17]. Suose the real index is fixed. The L Minkowski roblem for olytoes asks for the necessary and sufficient conditions on a set of unit vectors u 1,..., u n S d 1 and a set of real numbers α 1,..., α n > 0 that guarantee the existence of a olytoe, P, in R d containing the origin in its interior with n facets whose outer unit normals are u 1,..., u n S d 1 and such that if the facet with outer unit normal u i has area a i and distance from the origin h i, then for all i, h 1 i a i = α i Mathematics Subject Classification. 52A40. Research suorted, in art, by NSF Grant DMS

2 2 DANIEL HUG, ERWIN LUTWAK, DEANE YANG, AND GAOYONG ZHANG Obviously, the case = 1 is the classical roblem. For > 1 uniqueness was established in [11]. The L Minkowski roblem for olytoes is the discrete-data case of the general L Minkowski roblem (described below). In the discrete even-data case of the roblem, outer unit normals u 1, u 1,..., u m, u m are given in antiodal airs, where u i = u i, and α i = α i. With the excetion of the case = d, existence (and uniqueness) for the even roblem was established in [11] for all cases where the unit vectors do not lie in a closed hemishere of S d 1. A normalized version (discussed below) of the roblem was roosed and comletely solved for > 1 and even data in [18]. For d = 2, the imortant case = 0 of the discrete-data L Minkowski roblem was dealt with by Stancu [24], [25]. A solution to the L Minkowski roblem for > d was given by Guan and Lin [8] and indeendently by Chou and Wang [5]. The work of Chou and Wang [5] goes further and solves the roblem for olytoes for all > 1. The works of Guan and Lin [8] and Chou and Wang [5] focus on existence and regularity for the L Minkowski roblem. Both works make use of the machinery of the theory of PDE s. The classical Minkowski roblem has roven to be of interest to those working in both discrete and comutational geometry. It is likely that the L extension of the roblem will in time rove to be of interest to those working in these fields as well. An aroach accessible to researchers in convex, discrete, and comutational geometry aears to be desirable. This article resents two such aroaches. We begin by recalling the formulation of the L Minkowski roblem in full generality. For a convex body K let h K = h(k, ) : R d R denote the suort function of K; i.e., for x R d, let h K (x) = max y K x, y, where x, y is the standard inner roduct of x and y in R d. We shall use V (K) to denote d-dimensional volume of a convex body K in R d. The surface area measure, S(K, ), of the convex body K is a Borel measure on the unit shere, S d 1, such that V (K + εq) V (K) lim ε 0 + ε = h Q (u) S(K, du), S d 1 for each convex body Q. Here K + εq is the Minkowski combination defined by h(k + εq, ) = h(k, ) + εh(q, ). Existence of the surface area measure was shown by Aleksandrov and Fenchel and Jessen (see Schneider [21]). The classical Minkowski roblem asks for necessary and sufficient conditions for a Borel measure µ on S d 1 (called the data) to be the surface area measure of a convex body K. The solution as obtained by Aleksandrov and Fenchel and Jessen (see Schneider [21]) is: Corresonding to each Borel measure µ on S d 1 that is not concentrated on a closed hemishere of S d 1, there is a convex body K such that S(K, ) = µ if and only if S d 1 u dµ(u) = 0.

3 ON THE L MINKOWSKI PROBLEM FOR POLYTOPES 3 The uniqueness of K (u to translation) is a direct consequence of the Minkowski mixedvolume inequality (see Schneider [21]) which states that for convex bodies K, L, V (K + εq) V (K) lim dv (K) (d 1)/d V (L) 1/d, ε 0 + ε with equality if and only if K is a dilate of L (after a suitable translation). Suose > 1 is fixed and K is a convex body that contains the origin in its interior. The L surface area measure, S (K, ), of K is a Borel measure on S d 1 such that V (K + lim ε Q) V (K) = 1 h ε 0 + Q ε (u) S (K, du), S d 1 for each convex body Q that contains the origin in its interior. Here K + ε Q is the Minkowski-Firey L combination defined by h(k + ε Q, ) = h(k, ) + εh(q, ). Existence of the L surface area measure was shown in [11] where it was also shown that S (K, ) = h 1 K S(K, ). It is easily seen that the surface area measure of a convex body (and hence also all the L surface area measures) cannot be concentrated on a closed hemishere of S d 1. It turns out that if P is a olytoe with outer unit facet normals u 1,..., u n, then {u 1,..., u n } is the suort of the measure S(P, ) and S(P, {u i }) = a i where as before a i denotes the area of the facet of P whose outer unit normal is u i. Thus, if P contains the origin in its interior, S (P, {u i }) = h 1 i a i, where as before h i = h(p, u i ). The L Minkowski roblem asks for necessary and sufficient conditions for a Borel measure µ on S d 1 (called the data for the roblem) to be the L surface area measure of a convex body K; i.e., given a Borel measure µ on S d 1 that is not concentrated on a closed hemishere of S d 1, what are the necessary and sufficient conditions for the existence of a convex body K that contains the origin in its interior such that S (K, ) = µ or equivalently, h 1 K S(K, ) = µ. The roblem is of interest for all real. For > 1, but d, the uniqueness of K is a direct consequence of the L Minkowski mixed-volume inequality (established in [11]) which states that if > 1 then for convex bodies K, L, that contain the origin in their interior V (K + lim ε Q) V (K) d ε 0 + ε V (K)(d )/d V (L) /d, with equality if and only if K is a dilate of L. In [11] it was shown that if µ is an even Borel measure (i.e., assumes the same values on antiodal Borel sets) that is not concentrated on a great subshere of S d 1, then for each > 1, there exists a unique convex body K, that is symmetric about the origin such that S (K, ) = µ,

4 4 DANIEL HUG, ERWIN LUTWAK, DEANE YANG, AND GAOYONG ZHANG rovided d. The L Minkowski roblem as originally formulated cannot be solved for all even measures when = d. The following normalized version of the L Minkowski roblem was formulated in [18]: What are the necessary and sufficient conditions on a Borel measure µ to guarantee the existence of a convex body K, containing the origin in its interior, such that 1 V (K ) S (K, ) = µ? For all real d the two versions of the roblems are equivalent in that or equivalently K = V (K ) 1/( d) K K = V (K ) 1/ K. It was shown in [18] that the normalized L Minkowski roblem has a solution for all > 1 if the data measure is even (again assuming the measure is not concentrated on a subshere of S d 1 ). It is the aim of this note to resent two alternate aroaches to the Minkowski roblem which show that when the data is a discrete measure, the normalized version of the L Minkowski roblem always has a solution (assuming, as usual, that the measure is not concentrated on a closed hemishere of S d 1 ). It is imortant to emhasize that all of our results for > d were first obtained by Guan and Lin [8] and indeendently by Chou and Wang [5], and our results for > 1, were first obtained by Chou and Wang [5]. The sole aim of our work is to resent aroaches easily accessible to the convex, discrete, and comutational geometry community. 1. Results Let K d denote the sace of comact convex subsets of R d with nonemty interiors, and let P d denote the subset of convex olytoes. The members of K d are called convex bodies. We write K0 d for the set of convex bodies which contain the origin as an interior oint, and ut P0 d := P d K0. d For K K d, let F (K, u) denote the suort set of K with exterior unit normal vector u, i.e. F (K, u) = {x K : x, u = h(k, u)}. The (d 1)-dimensional suort sets of a olytoe P P d are called the facets of P. If P P d has facets F (P, u i ) with areas a i, i = 1,..., n, then S(P, ) is the discrete measure S(P, ) = a i δ ui with (finite) suort {u 1,..., u n } and S(P, {u i }) = a i, i = 1,..., n; here δ ui denotes the robability measure with unit oint mass at u i. Just as the L surface area measure of a convex body K K0 d satisfies S (K, ) = h(k, ) 1 S(K, ), the normalized L surface area measure of K is defined by S (K, ) := h(k, )1 S(K, ). V (K)

5 ON THE L MINKOWSKI PROBLEM FOR POLYTOPES 5 A convex body K is uniquely determined by its L surface area measure if > 1 and d (for = d one has uniqueness u to a dilatation), uniqueness holds for the normalized L surface area measure and all > 1. Again for a olytoe P P0 d with outer unit facet normals u 1,..., u n and facet areas a 1,..., a n > 0, i = 1,..., n, the discrete measures S (P, ) and S(P, ) are given by and S (P, ) = S (P, ) = h(p, u i ) 1 a i δ ui h(p, u i ) 1 a i δ ui. V (P ) In the case of a discrete measure µ = n j=1 α jδ uj with unit vectors u 1,..., u n not contained in a closed hemishere and α 1,..., α n > 0, any solution of the L Minkowski roblem for the data µ is necessarily a olytoe with facet normals u 1,..., u n (cf. [21, Theorem 4.6.4]). The main ste in our aroach to the L Minkowski roblem for general measures and general convex bodies is to solve first the L Minkowski roblem for discrete measures and olytoes. Theorem 1.1. Let vectors u 1,..., u n S d 1 that are not contained in a closed hemishere and real numbers α 1,..., α n > 0 be given. Then, for any > 1, there exists a unique olytoe P P0 d such that h(p, )1 α j δ uj = S(P, ). V (P ) j=1 From Theorem 1.1, we deduce the corresonding result for the L Minkowski roblem involving discrete measures and olytoes. Theorem 1.2. Let vectors u 1,..., u n S d 1 that are not contained in a closed hemishere and real numbers α 1,..., α n > 0 be given. Then, for any > 1 with d, there exists a unique olytoe P P0 d such that α j δ uj = h(p, ) 1 S(P, ). j=1 The extension of Theorem 1.1 to general measures will be obtained by aroximation with discrete measures. For each aroximating discrete measure, we get a olytoe as the solution of the discrete L Minkowski roblem. Then we show that a subsequence of these olytoes converges. However, the limit body may have the origin in its boundary. For this reason we are forced to slightly modify the original roblem. For d, we finally show by an additional argument that the original roblem is solved as well. Theorem 1.3. Let µ be a Borel measure on S d 1 whose suort is not contained in a closed hemishere of S d 1. Then, for any > 1, there exists a unique convex body K K d with 0 K such that V (K)h(K, ) 1 µ = S(K, ); moreover, K K d 0 if d.

6 6 DANIEL HUG, ERWIN LUTWAK, DEANE YANG, AND GAOYONG ZHANG Theorem 1.4. Let µ be a Borel measure on S d 1 whose suort is not contained in a closed hemishere of S d 1. Then, for any > 1 with d, there exists a unique convex body K K d with 0 K such that h(k, ) 1 µ = S(K, ); moreover, K K d 0 if > d. 2. Volume and diameter bounds The following three lemmas will be alied in two different ways. On the one hand, we will need them for our first treatment of the L Minkowski roblem for discrete measures and olytoes which is based on Aleksandrov s maing lemma (cf. [1]). Here the lemmas are alied in the very secial situation where all convex bodies are olytoes containing the origin in their interiors and with the same set of outer unit facet normals and where all measures are discrete with common finite suort. Excet for Lemma 2.1, the roofs of the lemmas in this secial case will not be simler than the ones in the general case. Therefore we resent them in the general framework. Then again Lemmas will be required for the solution of the L Minkowski roblem in the case of general convex bodies via an aroximation argument. The next lemma rovides a uniqueness result which will be used to establish the injectivity of an auxiliary ma (cf. Lemma 3.1) in our first roof of Theorem 1.1. It also yields the uniqueness assertions of Theorems 1.1 and 1.3. Moreover, an estimate established in the course of the roof of Lemma 2.1 will be emloyed in the roof of Lemma 2.2. Lemma 2.1. Let K, K K d be convex bodies with 0 K, K. Assume that µ is a Borel measure on S d 1 such that V (K)h(K, ) 1 µ = S(K, ) and V (K )h(k, ) 1 µ = S(K, ). Then K = K. Proof. Let L K d with 0 L. Define Ω := {u S d 1 : h(k, u) > 0} and Ω c := S d 1 \Ω. Then Hölder s inequality and the assumtion > 1 yield that ( ) 1 ( ( ) 1 h(l, u) ) 1 h(l, u) h(k, u)s(k, du) µ(du) d S d 1 Ω h(k, u) dv (K) h(l, u) h(k, u)s(k, du) Ω h(k, u) dv (K) (1) = V 1(K, L) V (K), since 1 h(k, u)s(k, du) = 1 h(k, u)s(k, du) = V (K) d Ω d S d 1 and S(K, Ω c ) = V (K) h(k, u) 1 µ(du) = 0. Ω c For L = K or L = K the left-hand side of (1) is equal to 1. Hence (1) and Minkowski s inequality (see [21, Theorem 6.2.1]) imly that 1 V 1(K, K ) V (K) ( ) V (K 1 ) d, V (K)

7 ON THE L MINKOWSKI PROBLEM FOR POLYTOPES 7 and therefore V (K) V (K ). By symmetry, we also have V (K) = V (K ), and thus K = K + t for some t R d. The assumtion and the translation invariance of the surface area measure now yield that [ h(k + t, u) 1 h(k, u) 1] µ(du) = 0 U for all Borel sets U S d 1. In articular, we may choose U t := {u S d 1 : t, u > 0}. If t 0, then U t is an oen hemishere. Since the suort of µ is not contained in S d 1 \ U t, we thus get ] [(h(k, u) + t, u ) 1 h(k, u) 1 µ(du) > 0. This shows that necessarily t = 0. U t In the following two lemmas we rovide a riori bounds for the volume and the diameter of solutions of the L Minkowski roblem. Lemma 2.2. Let µ be a ositive Borel measure on S d 1, and let K K d with 0 K satisfy V (K)h(K, ) 1 µ = S(K, ). Then ( ) d d V (K) κ d. µ(s d 1 ) Proof. Aly (1) with L = B d and use Minkowski s inequality (i.e. the isoerimetric inequality in this case) to get ( ) 1 ( ) 1 1 κd d d µ(sd 1 ), V (K) which is equivalent to the assertion of the lemma. Subsequently, we set α + := max{0, α} for α R. Let B d (0, r) denote the ball with center 0 and radius r 0. Lemma 2.3. Let µ and K be given as in Lemma 2.2. Assume that for some constant c 0 > 0, u, v + µ(du) d S c for all v S d 1. d 1 0 Then K B d (0, c 0 ). Proof. Define R := max{h(k, v) : v S d 1 } and choose v 0 S d 1 so that R = h(k, v 0 ). Then R[0, v 0 ] K, and thus R u, v 0 + h(k, u) for u S d 1. Hence R c R 1 u, v 0 + µ(du) 1 h(k, u) µ(du) 0 d S d d 1 S d 1 = 1 h(k, u)h(k, u) 1 µ(du) d S d 1 1 = h(k, u)s(k, du) = 1, dv (K) S d 1 which gives R c 0.

8 8 DANIEL HUG, ERWIN LUTWAK, DEANE YANG, AND GAOYONG ZHANG 3. The L Minkowski roblem for olytoes In this section, we will describe two different aroaches to Theorem 1.1. The first roof is based on the following auxiliary result, which is a minor modification of Aleksandrov s maing lemma. We include the roof for the sake of comleteness. Note that Aleksandrov used his maing lemma to solve the classical Minkowski roblem for olytoes. Lemma 3.1. Let A, B R n be nonemty oen sets, let B be connected, and let ϕ : A B be an injective, continuous ma. Assume that any sequence (x i ) i N in A with ϕ(x i ) b B as i has a convergent subsequence. Then ϕ is surjective. Proof. Since ϕ(a) B is nonemty, it is sufficient to show that ϕ(a) is oen and closed in B. Let b i ϕ(a), i N, with b i b B as i be given. Then there are x i A such that ϕ(x i ) = b i for i N. By assumtion, there is a subsequence (x i j ) j N with x i j x A as j. Since ϕ is continuous, ϕ(x i j ) ϕ(x) and therefore b = ϕ(x). Hence ϕ(a) is closed in B. Since A is oen in R n and ϕ is continuous and injective, ϕ(a) is oen in B by the theorem of the invariance of domain (cf. [20, Theorem 36.5] or [6, Theorem 4.3]). In the following, we write H u,t := {y R d : y, u t} for the halfsace with (exterior) normal vector u S d 1 and distance t 0 from the origin. For our first roof of Theorem 1.1, we can assume that the given vectors u 1,..., u n are airwise distinct and not contained in a closed hemishere. Let R n + be the set of all x = (x 1,..., x n ) R n with ositive comonents. For x R n +, we define the (comact, convex) olytoe n P (x) := Hu j,x j. j=1 The comactness of P (x) is imlied by the assumtion that u 1,..., u n are not contained in a closed hemishere. Since x R n +, 0 is an interior oint of P (x). Further, we remark that x P (x), x R n +, is continuous with resect to the Hausdorff metric (cf. [21,. 57]). We ut B := R n + and define A := {x R n + : S(P (x), {u j }) > 0 for j = 1,..., n}. Note that if x A, then x j = h(p (x), u j ) for j = 1,..., n. Clearly, A, B are nonemty oen subsets of R n and B is connected. Next we define the ma ϕ : A B by ϕ(x) := b = (b 1,..., b n ) with b j := h(p (x), u j) 1 S(P (x), {u j }) = S V (P (x)) (P (x), {u j }), j = 1,..., n. We will show that ϕ satisfies the assumtions of Lemma 3.1 to conclude that ϕ is surjective. The ma ϕ is well-defined and continuous. The continuity of ϕ follows from the continuity of the volume and the suort function and from the weak continuity of the surface area measure, since x P (x) is continuous as well. Next we check that ϕ is injective. Let x, y A be such that ϕ(x) = ϕ(y). Then Lemma 2.1 yields that P (x) = P (y). Hence, by the definition of A, x j = h(p (x), u j ) = h(p (y), u j ) = y j for j = 1,..., n, and thus x = y.

9 ON THE L MINKOWSKI PROBLEM FOR POLYTOPES 9 Now let x i A, i N, be given. Assume that b i := ϕ(x i ) b B as i and ut µ i := S (P (x i ), ) for i N. Since µ i (S d 1 ) = µ i ({u j }) = j=1 b i j j=1 j=1 b j as i, we get that µ i (S d 1 ) c 1 < for all i N. Hence, by Lemma 2.2 there is a constant c 2 > 0 such that, for i N, (2) V (P (x i )) c 2 > 0. For the discrete measure µ := n j=1 b jδ uj we have µ i µ weakly as i. The functions f i, f defined by f i (v) := u, v +µ i (dv), S d 1 f(v) := u, v +µ(dv), S d 1 v S d 1, are continuous and ositive since the suort of µ i, µ is not contained in a closed hemishere. Since f i converges uniformly to f as i and the shere is comact, there is a constant c 3 > 0 such that f i (v) c 3 for all v S d 1 and i N. Lemma 2.3 now imlies that there is a constant c 4 such that, for i N, (3) P (x i ) B d (0, c 4 ). By (3) there exists a convergent subsequence of P (x i ), i N. To simlify the notation, we assume that P (x i ) P P d as i. Note that by (2) P has indeed nonemty interior. Clearly, 0 P and the facets of P are among the suort sets F (P, u 1 ),..., F (P, u n ) of P with normal vectors u 1,..., u n. We next show that 0 int(p ). For this, assume that 0 is a boundary oint of P. Then there is a facet F (P, u j ) of P with 0 F (P, u j ) and S(P, {u j }) > 0, and therefore h(p, u j ) = 0. But then h(p (x i ), u j ) 0 and S(P (x i ), {u j }) 0, as i. In view of (3) this imlies that b i j = V (P (x i )) 1 S(P (xi ), {u j }) h(p (x i ), u j ) 1 as i, a contradiction. Since 0 int(p ), we get that h(p (x i ), u j ) 0 as i, for j = 1,..., n, and therefore also S(P (x i ), {u j }) 0; here we also use (2) and b i j b j 0 as i. This finally shows that S(P, {u j }) > 0 for j = 1,..., n. Thus we conclude that P = P (x) for x := (h(p, u 1 ),..., h(p, u n )) A and x i x as i. Now Lemma 3.1 shows that ϕ is surjective, which imlies the existence assertion of the theorem. Uniqueness has already been established in Lemma 2.1. We now give a second, variational roof of Theorem 1.1. An obvious advantage of this aroach is that it may be turned into a nonlinear reconstruction algorithm for retrieving a convex olytoe from its L surface area measure. The main difficulty consists in showing that the solution of an auxiliary otimization roblem is a convex olytoe which contains the origin in its interior.

10 10 DANIEL HUG, ERWIN LUTWAK, DEANE YANG, AND GAOYONG ZHANG The following lemma is robably well known. It will be used to verify that a convex olytoe which is defined as the solution of an auxiliary otimization roblem is indeed the solution of the L Minkowski roblem stated in Theorem 1.1. Lemma 3.2. Let u 1,..., u n S d 1 be airwise distinct vectors which are not contained in a closed hemishere. For x R n +, let P (x) := n H u i,x i and Ṽ (x) := V (P (x)). Then Ṽ is of class C 1 and i Ṽ (x) = S(P (x), {u i }) for i = 1,..., n. Proof. The second assertion can be checked by a direct argument. Alternatively, it can be obtained as a very secial case of Theorem in [21]. Here one has to choose Ω = {u 1,..., u n }, a ositive, continuous function f : S d 1 R with f(u j ) = x j, and a continuous function g i : S d 1 R with g i (u j ) = δ ij, for j = 1,..., n. The first assertion then follows, since x S(P (x), {u i }) is continuous on R n + (cf. the first roof of Theorem 1.1). We start with the second roof of Theorem 1.1. Again we can assume that u 1,..., u n are airwise distinct unit vectors not contained in a closed hemishere. Let α 1,... α n > 0 be fixed. We denote by R n the set of all x = (x 1,..., x n ) R n with nonnegative comonents. Then we define the comact set where M := {x R n : φ(x) = 1}, φ(x) := 1 d α i x i. For x M, we again write P (x) for the convex olytoe defined by n P (x) := Hu i,x i. Clearly, for any x M, 0 P (x) and P (x) has at most n facets whose outer unit normals are from the set {u 1,..., u n }. Moreover, h(p (x), u i ) x i with equality if S(P (x), {u i }) > 0, for i = 1,..., n. Since M is comact and the function x V (P (x)) =: Ṽ (x), x M, is continuous, there is a oint z M such that Ṽ (x) Ṽ (z) for all x M. We will rove that P (z) is the required olytoe. First, we show that (4) 0 int(p (z)). This will be roved by contradiction. Let h i := h(p (z), u i ) for i = 1,..., n. Without loss of generality, assume that h 1 =... = h m = 0 and h m+1,..., h n > 0 for some 1 m < n. Note that m < n is imlied by Ṽ (z) > 0. We will show that under this assumtion there is some z t M such that Ṽ (z t) > Ṽ (z), which contradicts the definition of z. Pick a small t > 0 and consider ( ( z t := (z 1 + t ) 1,..., (z m + t ) 1, z m+1 αt ) ) 1,..., (zn αt ) 1, where m α := α i n i=m+1 α. i Since 0 < h i z i for m + 1 i n, we have z t M if t > 0 is sufficiently small.

11 Define ON THE L MINKOWSKI PROBLEM FOR POLYTOPES 11 P t := m Hu i,t n i=m+1 H u i,(h i αt ) 1/, hence P 0 = P (z), P t P (z t ) and 0 int(p t ), if t > 0 is sufficiently small. We ut and thus and f i := S(P (z), {u i }) and i (t) := S(P t, {u i }) f i, dv (P t ) = t m (f i + i (t)) + dv 1 (P t, P (z)) = 0 i=m+1 m (f i + i (t)) + (h i αt ) 1 (fi + i (t)) i=m+1 h i (f i + i (t)) Since an interior oint of P (z) is also an interior oint of P t, if t > 0 is sufficiently small, it follows that P t P (z) as t 0 + (cf. [21,. 57]), and therefore i (t) 0 as t 0 +. From this and since at least one facet is suosed to contain the origin, we deduce that V (P t ) V 1 (P t, P (z)) lim t 0 + t ( m = 1 d lim t 0 + = 1 d m f i > 0. t 0 (f i + i (t)) + t i=m+1 ) (h i αt ) 1 hi (f i + i (t)) t Here the assumtion > 1 enters in a crucial way. By Minkowski s inequality and since P (t) P (z) as t 0 +, we get 0 < lim t 0 + V (P t ) V 1 (P t, P (z)) t lim inf t 0 + V (P t ) V (P t ) 1 1 d V (P (z)) 1 d = V (P (z)) 1 1 V (P t ) 1 d V (P (z)) 1 d d lim inf. t 0 + t But this shows that V (P t ) > V (P (z)) if t > 0 is sufficiently small. Since P t P (z t ), the required contradiction follows. From (4) it follows that z M + := {x R n + : φ(x) = 1}, and Ṽ (x) Ṽ (z) for all x M +. Hence, by the Lagrange multilier rule there is some λ R such that Ṽ (z) = λ φ(z). The required differentiability of Ṽ is ensured by Lemma 3.2, and φ(z) 0 since z Rn + and α 1,..., α n > 0; moreover, f i = λ 1 d α iz 1 i, i = 1,..., n, t

12 12 DANIEL HUG, ERWIN LUTWAK, DEANE YANG, AND GAOYONG ZHANG and thus λ > 0, since f i > 0 for some i {1,..., n}. We deduce that f i > 0 and therefore h(p (z), u i ) = z i for all i = 1,..., n. Since φ(z) = 1, we obtain that dv (P (z)) = This shows that, for i = 1,..., n, f i z i = λ 1 d α i z i = λ. S(P (z), {u i }) = f i = d V (P (z)) d α iz 1 i = V (P (z))h(p (z), u i ) 1 α i > The general case We now rovide a roof of Theorem 1.3. Let µ be a Borel measure on S d 1 whose suort is not contained in a closed hemishere. As in [21, ], one can construct a sequence of discrete measures µ i, i N, such that the suort of µ i is not contained in a closed hemishere and µ i µ weakly as i. By Theorem 1.1, for each i N there exists a olytoe P i P d 0 with µ i = h(p i, ) 1 S(P i, ). V (P i ) As in the roof of (3), we now obtain that the sequence P i, i N, is uniformly bounded. Hence we can assume that P i K K d as i and 0 K. In fact, since µ i (S d 1 ) µ(s d 1 ) as i, we get as in the roof of (2) that V (K) > 0, and thus K K d. For a continuous function f C(S d 1 ) and i N we have (5) f(u)v (P i )h(p i, u) 1 µ i (du) = S d 1 f(u)s(p i, du). S d 1 Since V (P i )h(p i, ) 1 V (K)h(K, ) 1 uniformly on S d 1 (note that 1 > 0), and since µ i µ and S(P i, ) S(K, ) weakly, as i, we obtain from (5) that (6) f(u)v (K)h(K, u) 1 µ(du) = S d 1 f(u)s(k, du). S d 1 The existence assertion now follows, since (6) holds for any f C(S d 1 ). Uniqueness has been roved in Lemma 2.1. Now we consider the case d. Assume that K K d with 0 K satisfies V (K)h(K, ) 1 µ = S(K, ), but 0 K. We will derive a contradiction by adating an argument from [5]. Let e S d 1 be such that K can locally be reresented as the grah of a convex function over B r := e B d (0, r), r > 0, and K H e,0 (cf. [2, Theorem 1.12]). Let µ i and P i P d 0 be constructed for µ as in the first art of the roof. In articular, µ i (S d 1 ) c 5 < and 0 int(p i ), for all i N, and P i K as i with resect to the Hausdorff metric. Then, for i i 0, P i can locally be reresented as the grah of a convex function g i over B r, and the Lischitz constants of these functions are uniformly bounded by some constant L. We define G i (y) := y + g i (y)e for y B r, ut α := 1 and write c 6, c 7, c 8 for constants

13 ON THE L MINKOWSKI PROBLEM FOR POLYTOPES 13 indeendent of i and r. Then, for i i 0, c 5 µ i (S d 1 1 ) = h(p i, u) α S(P i, du) V (P i ) S d 1 c 6 x, σ(p i, x) α H d 1 (dx), G i (B r) where H d 1 denotes the (d 1)-dimensional Hausdorff measure and σ(p i, x) is an exterior unit normal vector of P i at x P i, which is uniquely determined for H d 1 -almost all x P i. Using the area formula and the fact that for H d 1 -almost all y B r, we obtain Since we further deduce that σ(p i, G i (y)) = ( 1 + g i (y) 2) 1 2 ( g i (y) e), c 5 c 6 B r G i (y), σ(p i, G i (y)) α 1 + g i (y) 2 H d 1 (dy) = c 6 B r ( y, g i (y) g i (y)) α 1 + g i (y) 21 α H d 1 (dy) c 7 B r ( y, g i (y) g i (y)) α H d 1 (dy). 0 < y, g i (y) g i (y) 2L y + g i (0), c 5 c 7 B r (2L y + g i (0) ) α H d 1 (dy) = c 8 r 0 (2Lt + g i (0) ) α t d 2 dt. Since g i (0) 0 as i, we can extract a decreasing subsequence of ( g i (0) ) i N. Hence the monotone convergence theorem yields that r c 5 c 8 (2Lt) α t d 2 dt. This leads to a contradiction if α d 1, since r > 0 can be arbitrarily small. 0 Examle 4.1. We now give an examle of a Borel measure µ on S d 1 whose suort is not contained in a hemishere such that 0 is a boundary oint of the uniquely determined convex body K K d for which V (K)h(K, ) 1 µ = S(K, ). For q > 1 we define g(x) := x q for x R d 1 and K := {(x, t) R d 1 R : t g(x)} H e d,1. Clearly, K K d, 0 K and K is C 2 in a neighbourhood of 0 excluding 0. The given convex body satisfies V (K)h(K, ) 1 µ = S(K, ) if µ := h(k, )1 S(K, ) V (K) defines a finite measure on S d 1 and S(K, { e d }) = 0. Since indeed S(K, { e d }) = 0 and h(k, u) > 0 for u S d 1 \ { e d }, and since S(K, ) is absolutely continuous with resect to the sherical Lebesgue measure (with density function f K ) in a sherical neighbourhood of

14 14 DANIEL HUG, ERWIN LUTWAK, DEANE YANG, AND GAOYONG ZHANG e d, it remains to show that h(k, ) 1 f K is integrable in a sherical neighbourhood of e d. For r (0, 1) we ut B r := B d (0, r) e d. Then we define a(x) := (1 + g(x) 2 ) 1/2, x B r, where g(x) = q x q 2 x. For x B r \ {0} and we get u := σ(k, (x, g(x))) = a(x) 1 ( g(x) e d ), h(k, u) = x + g(x)e d, u = a(x) 1 (q 1) x q, f K (u) 1 = a(x) (d+1) det ( d 2 g(x) ), and hence h(k, u) 1 f K (u) = (q 1) 1 a(x) d+ x [ q(1 ) det ( d 2 g(x) )] 1. A direct comutation shows that and therefore det ( d 2 g(x) ) = q d 1 (q 1) x (q 2)(d 1), h(k, u) 1 f K (u) = q 1 d (q 1) x [(q 2)(d 1)+q( 1)] a(x) d+ For a given (1, d), we now choose 2(d 1) q := (1, 2), d + 2 and hence h(k, u) 1 f K (u) = q 1 d (q 1) a(x) d+. Since x a(x) is bounded on B r, the required integrability follows. A more recise estimate shows that h(k, ) 1 f K is integrable whenever q > 1 and < d 1 + q. q For q = 2, K is C 2 and has ositive curvature at 0 and h(k, ) 1 f K is integrable for 1 < < (d + 1)/2. References [1] A.D. Aleksandrov, Konvexe Polyeder, Akademie-Verlag, Berlin, (Russian original: 1950), [2] H. Busemann, Convex Surfaces, Interscience, New York, [3] S. Cami, P. Gronchi, The L -Busemann-Petty centroid inequality, Adv. Math. 167 (2002), [4] S. Cami, P. Gronchi, On the reverse L -Busemann-Petty centroid inequality, Mathematika (in ress). [5] K.-S. Chou, X.-J. Wang, The L -Minkowski roblem and the Minkowski roblem in centroaffine geometry, Prerint, July [6] K. Deimling, Nonlinear Functional Analysis, Sringer, Berlin, [7] R. J. Gardner, Geometric Tomograhy, Cambridge Univ. Press, Cambridge, [8] P. Guan, C-S. Lin, On equation det(u ij + δ ij u) = u f on S n, rerint. [9] D. Hug and R. Schneider, Stability results involving surface area measures of convex bodies, Rend. Circ. Mat. Palermo (2) Sul. No. 70, art II (2002), [10] M. Ludwig, Ellisoids and matrix valued valuations, Duke Math J. 119 (2003), [11] E. Lutwak, The Brunn-Minkowski-Firey theory I: mixed volumes and the Minkowski roblem, J. Differential Geom. 38 (1993), [12] E. Lutwak, The Brunn-Minkowski-Firey theory II: Affine and geominimal surface areas, Adv. Math. 118 (1996),

15 ON THE L MINKOWSKI PROBLEM FOR POLYTOPES 15 [13] E. Lutwak, V. Oliker, On the regularity of solutions to a generalization of the Minkowski roblem, J. Differential Geom. 41 (1995), [14] E. Lutwak, D. Yang, and G. Zhang, A new ellisoid associated with convex bodies, Duke Math. J. 104 (2000), [15] E. Lutwak, D. Yang, and G. Zhang, L affine isoerimetric inequalities, J. Differential Geom. 56 (2000), [16] E. Lutwak, D. Yang, and G. Zhang, The Cramer-Rao inequality for star bodies, Duke Math. J. 112 (2002), [17] E. Lutwak, D. Yang, and G. Zhang, Shar affine L Sobolev inequalities, J. Differential Geom. 62 (2002), [18] E. Lutwak, D. Yang, and G. Zhang, On the L -Minkowski roblem, Trans. Amer. Math. Soc. (to aear),. 15. [19] M. Meyer, E. Werner, On the -affine surface area, Adv. Math. 152 (2000), [20] J.R. Munkres, Elements of Algebraic Toology, Addison-Wesley, Menlo Park, California, [21] R. Schneider, Convex Bodies: the Brunn-Minkowski Theory, Encycloedia of Mathematics and its Alications 44, Cambridge University Press, Cambridge, [22] C. Schütt, E. Werner, Polytoes with vertices chosen randomly from the boundary of a convex body Sringer Lecture Notes in Mathematics 1807 (2003), [23] C. Schütt, E. Werner, Surface bodies and -affine surface area, Adv. Math. (in ress). [24] A. Stancu, The discrete lanar L 0 -Minkowski roblem, Adv. Math. 167 (2002), [25] A. Stancu, On the number of solutions to the discrete two-dimensional L 0 -Minkowski roblem, Adv. Math., in ress. [26] V. Umanskiy, On solvability of the two dimensional L -Minkowski roblem, Adv. Math. (in ress). [27] E. Werner, The -affine surface area and geometric interretations, Rend. Circ. Mat. Palermo (IV International Conference in Stochastic Geometry, Convex Bodies, Emirical Measures & Alications to Engineering Science, Troea, 2001) 70 (2002), Mathematisches Institut, Universität Freiburg, Eckerstr. 1, D Freiburg, Germany Deartment of Mathematics, Polytechnic University, Six Metrotech Center, Brooklyn, NY address: daniel.hug@math.uni-freiburg.de address: elutwak@oly.edu address: dyang@oly.edu address: gzhang@oly.edu