Extremum problems for the cone volume functional of convex polytopes

Lutwak, Yang and Zhang defined the cone volume functional U over convex polytopes in Rn containing the origin in their interiors, and conjectured that the greatest lower bound on the ratio of this centro-affine invariant U to volume V is attained by parallelotopes. In this paper, we give affirmative answers to the conjecture in R² and R³. Some new sharp inequalities characterizing parallelotopes in Rn are established. Moreover, a simplified proof for the conjecture restricted to the class of origin-symmetric convex polytopes in Rn is provided.     

On the minimal volume of simplices enclosing a convex body

Let \({C \subset \mathbb{R}^n}\) be a compact convex body. We prove that there exists an n-simplex \({S\subset \mathbb{R}^n}\) enclosing C such that \({{\rm Vol}(S) \leq n^{n-1} {\rm Vol}(C)}\) .     

Translative packing of a convex body by sequences of its positive homothetic copies

Every sequence of positive homothetic copies of a planar convex body C whose total area does not exceed a quarter of the area of C can be translatively packed in C.      

On the duality operator of a convex cone

Let C be a convex set in Rⁿ. For each y?C, the cone of C at y, denoted by cone(y, C), is the cone {α(x ? y): α ? 0 and x?C}. If K is a cone in Rⁿ, we shall denote by $\text{K}{}^1$ its dual cone and by F(K) the lattice of faces of K. Then the duality operator of K is the mapping $\text{d}{}_{\mathrm{K}}\text{: F(K) → F(K}{}^1\text{)}$ given by $\text{d}{}_{\mathrm{K}}\text{(F) = (}\text{span}\text{F)}{}^{\mathrm{\perp }}\text{∩ K}{}^1$. Properties of the duality operator d_K of a closed, pointed, full cone K have been studied before. In this paper, we study d_K for a general cone K, especially in relation to d_{cone(y, K)}, where y?K. Our main result says that, for any closed cone K in Rⁿ, the duality operator d_K is injective (surjective) if and only if the duality operator d_{cone(y, K)} is injective (surjective) for each vector y?K ～ [K ∩ (? K)]. In the last part of the paper, we obtain some partial results on the problem of constructing a compact convex set C, which contains the zero vector, such that cone (0, C) is equal to a given cone.     

Capillary surfaces in a convex cone

We show that a compact embedded hypersurface ${S\subset\mathbb{R}^{n+1}}$ with constant higher-order mean curvature in a convex piecewise smooth cone C which is perpendicular to ?C is part of a hypersphere. Also we prove that an embedded disk type constant mean curvature surface ${S\subset\mathbb{R}^3}$ in a convex polyhedral cone C which makes constant contact angles with ?C is a spherical cap if C has at most five faces. This condition on the number of faces can be dropped if C is a right cone over a regular n-gon and the contact angles are the same on ?S.     

Random walks in a convex body and an improved volume algorithm

We give a randomized algorithm using O(n⁷ log² n) separation calls to approximate the volume of a convex body with a fixed relative error. The bound is O(n⁶ log⁴ n) for centrally symmpetric bodies and for polytopes with a polynomial number of facets, and O(n⁵ log⁴ n) for centrally symmetric polytopes with a polynomial number of facets. We also give an O(n⁶ log n) algorithm to sample a point from the uniform distribution over a convex body. Several tools are developed that may be interesting on their own. We extend results of Sinclair–Jerrum [43] and the authors [34] on the mixing rate of Markov chains from finite to arbitrary Markov chains. We also analyze the mixing rate of various random walks on convex bodies, in particular the random walk with steps from the uniform distribution over a unit ball. © 1993 John Wiley & Sons, Inc.     

The mean volume of boxes and cylinders circumscribed about a convex body

The mean volume of boxes circumscribed about a convex bodyK of given volume is a minimum whenK is a ball. This follows from a more general inequality, where the volume of circumscribed boxes is replaced by the product of quermassintegrals of the projections ofK on appropriate lower dimensional subspaces.     

Sets convex in a cone of directions

We consider sets which are convex in directions from some cone K. We generalize some well-known properties of ordinary convex sets for the case of K-convex sets and give some applications in optimization theory.     

Computing the radius of pointedness of a convex cone

Let Ξ(H) denote the set of all nonzero closed convex cones in a finite dimensional Hilbert space H. Consider this set equipped with the bounded Pompeiu-Hausdorff metric δ. The collection of all pointed cones forms an open set in the metric space (Ξ(H),δ). One possible way of measuring the degree of pointedness of a cone K is by evaluating the distance from K to the set of all nonpointed cones. The number ρ(K) obtained in this way is called the radius of pointedness of the cone K. The evaluation of this number is, in general, a very cumbersome task. In this note, we derive a simple formula for computing ρ(K), and we propose also a method for constructing a nonpointed cone at minimal distance from K. Our results apply to any cone K whose maximal angle does not exceed 120°. Dedicated to Clovis Gonzaga on the occassion of his 60th birthday.     

How to hold a convex body?

Convex bodies are often used for mathematical tests. They occasionally try to escape. Can the testing mathematician hold them still by using a circle? Rarely not.     

Constructing a polytope to approximate a convex body

We develop an algorithm to construct a convex polytopeP withn vertices, contained in an arbitrary convex bodyK inR ^d , so that the ratio of the volumes |K/P|/|K| is dominated byc ·. d/n ^2/(d–1).Supported in part by the fund for the promotion of research in the Technion     

On the minimal convex annulus of a planar convex body

In this paper we introduce the notion of a minimal convex annulusK (C) of a convex bodyC, generalizing the concept of a minimal circular annulus. Then we prove the existence — as for the minimal circular annulus — of a Radon partition of the set of contact points of the boundaries ofK (C) andC. Subsequently, the uniqueness ofK (C) is shown. Finally, it is proven that, for typicalC, the boundary ofC has precisely two points in common with each component of the boundary ofK (C).     

On the inner parallel body of a convex body

It is shown that the set of boundary points of a convex body at which there are no interior tangent balls of positive radius has zero surface area.     

Stein's phenomenon in estimation of means restricted to a polyhedral convex cone

This paper treats the problem of estimating the restricted means of normal distributions with a known variance, where the means are restricted to a polyhedral convex cone which includes various restrictions such as positive orthant, simple order, tree order and umbrella order restrictions. In the context of the simultaneous estimation of the restricted means, it is of great interest to investigate decision-theoretic properties of the generalized Bayes estimator against the uniform prior distribution over the polyhedral convex cone. In this paper, the generalized Bayes estimator is shown to be minimax. It is also proved that it is admissible in the one- or two-dimensional case, but is improved on by a shrinkage estimator in the three- or more-dimensional case. This means that the so-called Stein phenomenon on the minimax generalized Bayes estimator can be extended to the case where the means are restricted to the polyhedral convex cone. The risk behaviors of the estimators are investigated through Monte Carlo simulation, and it is revealed that the shrinkage estimator has a substantial risk reduction.