Existence Of Convex Body Research Articles

Application of an idea of Voronoĭ to lattice packing, supplement

Voronoĭ’s criterion on lattice packings of balls in \(\mathbb E^d\) of local maximum density was extended in an earlier article to lattice packings of convex bodies in \(\mathbb E^d\) with local ultra maximum density, a stronger version of maximum density. No example of such a packing was given. However, it was conjectured that for generic convex bodies, the lattice packings of local maximum density are of local ultra maximum density provided that the dimension \(d\) is sufficiently high. For generic convex bodies, the kissing number of lattice packings of global maximum density is between \(d(d+1)\) and \(2d^2\). We conjecture that it equals \(2d^2\). In this note, first steps toward these conjectures are made. The existence of convex bodies with lattice packings of local ultra maximum density is shown. Contrasting this, families of convex bodies are specified which do not admit lattice packings of local ultra maximum density. If for a dense family of convex bodies every lattice packing of global maximum density has ultra maximum density, then this holds for a dense open family of convex bodies as well. A related result treats kissing numbers.

Two problems related to prescribed curvature measures

Existence of convex body with prescribed generalized curvaturemeasures is discussed, this result is obtained by making use ofGuan-Li-Li's innovative techniques. Moreover, we promote Ivochkina's$C^2$ estimates for prescribed curvature equation in [12,13].

The Existence of Convex Body with Prescribed Curvature Measures

Curvature measure and surface area measure are the basic notions in the classical differential geometry. They play fundamental roles in the theory of convex bodies. They are closely related to the differential geometry and integral geometry of convex hypersurfaces. The Minkowski problem is the problem of prescribing n-th surface area measure on Sn. The Christoffel problem concerns the prescribing the 1-st surface area measure (e.g., see [1, 14, 17, 6, 19, 7, 3]). The general problem of prescribing surface area measures is called the Christoffel-Minkowski problem, we refer [12] for an updated account. The problem of prescribing 0-th curvature measure is called the Alexandrov problem, which is a counterpart to Minkowski problem. The problem is equivalent to solve a Monge-Ampere type equation on Sn. The existence and uniqueness were obtained by Alexandrov [2]. The regularity of the Alexandrov problem in elliptic case was proved by Pogorelov [18] for n = 2 and by Oliker [16] for higher dimension case. The general regularity results (degenerate case) of the problem were obtained in [9]. The general problem of prescribing (n− k)-th curvature measure for case k ≤ n is an interesting counterpart of the Christoffel-Minkowski problem. It has been discussed in literature (e.g., [20]). Nevertheless, very little is known except for the Alexandrov problem. In this paper, we are concerned with the existence of convex bodies with the prescribed (n− k)-th curvature measure for 1 ≤ k < n. We start with the definitions of curvature measures and surface area measures for convex bodies with smooth boundary. Let Ω be a bounded convex body in Rn+1 with C2 boundary M , the corresponding curvature measures and surface area measures of Ω can be defined according to some geometric quantities of M . Let κ = (κ1, · · · , κn) be the principal curvatures of M at point x, let Wk(x) = Sk(κ(x)) be the k-th Weingarten curvature of M at x (where Sk is the k-th elementary symmetric function). In particular, W1,W2

On the Christoffel-Minkowski problem of Firey?s p-sum

The classical Brunn-Minkowski theory for convex bodies was developed from a few basic concepts: support functions, Minkowski combinations, and mixed volumes. As a special case of mixed volumes, the Quermassintegrals are important geometrical quantities of a convex body, and surface area measures are local versions of Quermassintegrals. The Christoffel-Minkowski problem concerns with the existence of convex bodies with prescribed surface area measure, for details please refer to [17, 3, 8, 18]. In 1962, Firey [5] generalized the Minkowski combination to p-sums from p = 1 to p ≥ 1. Later, Lutwak [13, 14] showed that Firey’s p-sum also leads to a Brunn-Minkowski theory for each p > 1. This theory has found many geometry applications, see for example, [16] and its references. It was also shown in [13] that the classical surface area measures could be extended to the p-sum case. So it is natural to consider a generalization of the classical Christoffel-Minkowski problem for each p > 1. The generalized Minkowski problem has been treated in [13, 15, 7, 4]. In this paper we study the remaining case, which may be called the Christoffel-Minkowski problem of p-sum. First we introduce some notations and relevant results. Let Kn+1 denote the set of convex bodies (compact, convex subsets with nonempty interiors) in Euclidean space Rn+1, and let Kn+1 0 denote the set of convex bodies containing the origin in their interiors. For K ∈ Kn+1, let hK = h(K, ·) : Sn → R denote the support function of K, and let W0(K),W1(K), ..., Wn+1(K) denote the Quermassintegrals of K (see, for example [18]). Thus W0(K) = V (K), the volume of K and Wn+1 = V (B) = ωn+1, where B is the unit ball in Rn+1. For each Firey p-sum, Lutwak [13] defined the mixed p-Quermassintegrals by n + 1− k p Wp,k(K, L) = lim 2→0+ Wk(K +p 2 · L)−Wk(K) 2 . (1.1)