On a geometric extremum problem for convex cones
Oleg Mushkarov, Nikolai Nikolov
TL;DR
This work analyzes the extremal geometry problem of minimizing the $(n-1)$-volume of the intersection of a pointed convex cone $K$ with a hyperplane through a fixed point $A$, under admissibility constraints. It develops a geometric stationarity criterion, showing that a hyperplane $\mathcal{H}$ is stationary iff $\overrightarrow{AH}=n\overrightarrow{AG}$, and characterizes stationary hyperplanes for hyperangles via equivalences involving the centroid and Monge point. Special attention is given to the non-negative orthant, where every $A\in K^o$ has a unique stationary hyperplane, explicitly described by a positive root of an irrational equation, and with a closed-form for the associated $(n-1)$-volume; in dimension two, the shortest cone-cut segment length is given by a sharp $a_1^{2/3}+a_2^{2/3}$ expression. These results illuminate how extremal hyperplane positions depend on cone geometry, yielding explicit constructions and length formulas in orthant settings and clarifying the difference between low and high dimensions ($n=2$ vs $n\ge3$).
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.