On a geometric extremum problem for convex cones

TL;DR

This paper investigates the geometric extremum problem of minimizing the intersection volume of convex cones with hyperplanes, providing characterizations of stationary hyperplanes and analyzing specific cases like the non-negative orthant.

Contribution

It offers a geometric characterization of stationary hyperplanes for convex cones and studies the set of points with such hyperplanes, including detailed analysis of the non-negative orthant.

Findings

Characterization of stationary hyperplanes for hyperangle cones

Description of the set of points with stationary hyperplanes

Analysis of cone segments in the non-negative orthant

Abstract

We discuss the optimization problem for minimizing the $(n-1)$-volume of the intersection of a convex cone $K$ in $\Bbb R^n$ with a hyperplane through a given point, first considered in \cite{We}. We give a geometric characterization of the stationary hyperplanes for this problem when $K$ is a hyperangle which partially answers a question posed in \cite{We}. Moreover, we study the location of the set $S$ of points for which there is a stationary hyperplane as well as the infimum of the $(n-1)$-volumes of cone segments of $K$ cut off by hyperplanes through a given boundary point of $K$. As a model example we study in detail the non-negative orthant of $\Bbb R^n$. In this case $S$ is its interior and we show that every point of $S$ lies in a unique stationary hyperplane, which we describe in terms of the unique real root of an irrational equation.