arXiv subjects

Steven Hoehner

Publications and source records attributed to Steven Hoehner.

22 records · Page 2Linked to original sources

arXiv2021

Extremal general affine surface areas

For a convex body $K$ in $\mathbb{R}^n$, we introduce and study the extremal general affine surface areas, defined by \[ {\rm IS}_φ(K):=\sup_{K^\prime\subset K}{\rm as}_φ(K),\quad {\rm os}_ψ(K):=\inf_{K^\prime\supset K}{\rm as}_ψ(K) \] where ${\rm as}_φ(K)$ and ${\rm as}_ψ(K)$ are the $L_φ$ and $L_ψ$ affine surface area of $K$, respectively. We prove that there exist extremal convex bodies that achieve the supremum and infimum, and that the functionals ${\rm IS}_φ$ and ${\rm os}_ψ$ are continuous. In our main results, we prove Blaschke-Santaló type inequalities and inverse Santaló type inequalities for the extremal general affine surface areas. This article may be regarded as an Orlicz extension of the recent work of Giladi, Huang, Schütt and Werner (2020), who introduced and studied the extremal $L_p$ affine surface areas.

math.FA↗

arXiv2020

The Maximum Surface Area Polyhedron with Five Vertices Inscribed in the Sphere $\mathbb{S}^2$

This article focuses on the problem of analytically determining the optimal placement of five points on the unit sphere $\mathbb{S}^2$ so that the surface area of the convex hull of the points is maximized. It is shown that the optimal polyhedron has a trigonal bipyramidal structure with two vertices placed at the north and south poles and the other three vertices forming an equilateral triangle inscribed in the equator. This result confirms a conjecture of Akkiraju, who conducted a numerical search for the maximizer. As an application to crystallography, the surface area discrepancy is considered as a measure of distortion between an observed coordination polyhedron and an ideal one. The main result yields a formula for the surface area discrepancy of any coordination polyhedron with five vertices.

math.MG↗

arXiv2020

A Concentration Inequality for Random Polytopes, Dirichlet-Voronoi Tiling Numbers and the Geometric Balls and Bins Problem

Our main contribution is a concentration inequality for the symmetric volume difference of a $ C^2 $ convex body with positive Gaussian curvature and a circumscribed random polytope with a restricted number of facets, for any probability measure on the boundary with a positive density function. We also show that the Dirichlet-Voronoi tiling numbers satisfy $ \text{div}_{n-1} = (2πe)^{-1}(n+\ln n) + O(1)$, which improves a classical result of Zador by a factor of $o(n)$. In addition, we provide a remarkable open problem which is the natural geometric generalization of the famous and fundamental "balls and bins" problem from probability. This problem is tightly connected to the optimality of random polytopes in high dimensions.

math.MG↗

arXiv2020

Intrinsic and Dual Volume Deviations of Convex Bodies and Polytopes

We establish estimates for the asymptotic best approximation of the Euclidean unit ball by polytopes under a notion of distance induced by the intrinsic volumes. We also introduce a notion of distance between convex bodies that is induced by the Wills functional, and apply it to derive asymptotically sharp bounds for approximating the ball in high dimensions. Remarkably, it turns out that there is a polytope which is almost optimal with respect to all intrinsic volumes simultaneously, up to absolute constants. Finally, we establish asymptotic formulas for the best approximation of smooth convex bodies by polytopes with respect to a distance induced by dual volumes, which originate from Lutwak's dual Brunn-Minkowski theory.

math.MG↗