Estimates for measures of lower dimensional sections of convex bodies

A generalized localization theorem and geometric inequalities for convex bodies

The dual Minkowski problem for symmetric convex bodies

Minkowski proved two important finiteness theorems concerning the reduction theory of positive definite quadratic forms (see [6], p. 285 or [7], §8 and §10). A positive definite quadratic form in n variables may be considered as an ellipsoid in n-dimensional Euclidean space, Rn, and then the two results can be investigated more generally by replacing the ellipsoid by any symmetric convex body in Rn. We show here that when n≧3 the two finiteness theorems hold only in the case of the ellipsoid. This is equivalent to showing that Minkowski's results do not hold in a general Minkowski space, namely in a euclidean space where the unit ball is a general symmetric convex body instead of the sphere or ellipsoid.

High dimensional random sections of isotropic convex bodies

A [formula omitted] estimate for measures of hyperplane sections of convex bodies

ON ANDERSON'S PROBABILITY INEQUALITY

In this paper, we consider the extremal problem of the lp-norm: min{lp(TK), ℴ∈TK⊆L,T∈GL(n)}, where K,L are two convex bodies in ℝn. Using the optimization theorem of John, we give necessary conditions for K to be in extremal position in terms of a decomposition of the identity. Furthermore, the weaker optimization problem, min{(lp(TK))p: TK ⊆ B2n, TK∩Sn−1 ≠ ∅, T∈GL(n)}, is also considered. As an application, the geometric distance between the unit ball B2n and a centrally symmetric convex body K is obtained.

In this paper, we will denote by convex figure a compact convex subset of the n-dimensional Euclidean space ℝn, and by convex body a convex figure with non-empty interior. The principal kinematic formula in integral geometry gives the measure of the set of congruent convex bodies intersecting with a fixed convex body. Specifically, let K, L be two convex bodies in ℝn and G(n) the group of special motions in ℝn. Each element, g: ℝn → ℝn, of G(n) can be represented bywhere b∈ℝn and e is an orthogonal matrix of determinant 1. Let μ be the Haar measure on G(n) normalized as follows: Let μ:ℝn × SO(n) → G(n) be defined by φ(t, e)x = ex + t, xeℝn, where SO(n) is the rotation group of ℝn. If v is the unique invariant probability measure on SO(n), η is the Lebesgue measure on ℝn, then μ is chosen as the pull back measure of η⊗v under φ−1. If Wi(K), Wi(L) are the quermassintegrals of K, L, i= 0, 1,…, n, the principal kinematic formula states thatwhere ωn is the volume of the unit n–ball.

1. Let A1, * , An be n linearly independent points in Rn, the n-dimensional Euclidean space. The set A= u= Ai+ * * * +unAn Ul, ** *, Un integers} is called a lattice, and {A1, * * *, AI is called a base of A. Let A i have co-ordinates ali, ...*, ani. Then d(A) = I det(ai,) j is called the determinant of the lattice A; it is independent of the choice of a base of A. Let S be a set in Rn. A lattice A is said to be S-admissible if A has no point other than the origin 0 in the interior of S. The critical determinant A(S) of S is defined by A(S) -inf d(A), where A runs over all S-admissible lattices (A(S) = oo if S has no admissible lattice). Clearly, SC T implies A(S) ?A(T). One of the principal problems in Geometry of Numbers is to find a method for determining A(S) for a given set S. For twoand threedimensional symmetrical' convex bodies, Minkowski reduced the problem to the discussion of special classes of lattices with points on the boundary of the body. This method has been partially extended to symmetrical convex bodies in R4, but it is very difficult to apply in spaces of dimension higher than two. Reinhardt [7] and Mahler [4] independently proved that for a symmetrical convex domain K in R2, A(K) =H(K)/4, where H(K) denotes the area of a smallest symmetrical hexagon2 containing K. In other words, a symmetrical convex domain can be inscribed in a space-filling symmetrical convex domain with the same critical determinant. The straightforward generalization of this result to higher dimensions would be: If K is a symmetrical convex body in Rn then K is contained in a space-filling symmetrical convex body (P with A(K) =A((P) =V(6)/2n, where V(Q?) denotes the volume of (P. The object of this note is to prove that this generalization does not hold in R,, for n _ 3.

For a convex body K C iR, the kth projection function of K assigns to any k-dimensional linear subspace of RI the k-volume of the or thogonal projection of K to that subspace. Let K and Ko be convex bodies in Rn, and let Ko be centrally symmetric and satisfy a weak regularity assump tion. Let i, j E N be such that 1 < i < j < n-2 with (i, j) :A (1, n-2). Assume that K and Ko have proportional ith projection functions and proportional jth projection functions. Then we show that K and Ko are homothetic. In the particular case where Ko is a Euclidean ball, we thus obtain characteri zations of Euclidean balls as convex bodies having constant i-brightness and constant j-brightness. This special case solves Nakajima's problem in arbitrary dimensions and for general convex bodies for most indices (i, j).

Hyperspaces of smooth convex bodies up to congruence

Extremizers in Soprunov and Zvavitch's Bezout inequalities for mixed volumes

Schneider introduced an inter-dimensional difference body operator on convex bodies, and proved an associated inequality. In the prequel to this work, we showed that this concept can be extended to a rich class of operators from convex geometry and proved the associated isoperimetric inequalities. The role of cosine-like operators, which generate convex bodies in Rn from those in Rn, were replaced by inter-dimensional simplicial operators, which generate convex bodies in Rnm from those in Rn (or vice versa). In this work, we treat the Lp extensions of these operators, and, furthermore, extend the role of the simplex to arbitrary m-dimensional convex bodies containing the origin. We establish mth-order Lp isoperimetric inequalities, including the mth-order versions of the Lp Petty projection inequality, Lp Busemann-Petty centroid inequality, Lp Santaló inequalities, and Lp affine Sobolev inequalities. As an application, we obtain isoperimetric inequalities for the volume of the operator norm of linear functionals (Rn,‖⋅‖E)→(Rm,‖⋅‖F).

In the first three sections of this chapter, we will investigate four affine invariant problems referring to convex bodies in Rn. It is shown that these problems are equivalent for compact, convex bodies, whereas they differ from each other in the unbounded case. Among these four problems, the central one is the question for the minimal number of smaller homothets of a convex body M ⊂ Rn which are sufficient to coverM. In addition, the problem of illuminating of the boundary bd M by the smallest number of directions is discussed. A lot of partial results regarding both the problems are known, but for n ≥ 3 the general solutions are still unknown. We give a survey on the contributions up to the recent state.KeywordsBoundary PointConvex BodySupporting LineSmall Positive IntegerOuter NormalThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Let and let and be two convex bodies in such that their orthogonal projections and onto any ‐dimensional subspace are directly congruent, that is, there exists a rotation and a vector such that . Assume also that the 2‐dimensional projections of and are pairwise different and they do not have ‐symmetries. Then and are congruent. We also prove an analogous more general result about twice differentiable functions on the unit sphere in .