Regions exhibiting stationary values of average cross section

TL;DR

This paper investigates shapes with stationary average cross sections, deriving formulas and analyzing properties across multiple dimensions to understand optimal geometric configurations.

Contribution

It introduces a new formula for average cross section in any dimension and characterizes stationary shapes through algebraic, numerical, and graphical methods.

Findings

Stationary shapes have specific algebraic properties.

Average cross section maximization requires cross sections passing through the centroid.

Stationary shape characteristics vary across dimensions 2, 3, and 5.

Abstract

The problem of maximizing the average cross section through a point within a shape is introduced. This idea is extended into arbitrary dimensions. However, the average cross sectional volume cannot be maximized unless the cross sections pass through the centroid of the shape. Therefore, we focus on the shapes with stationary values of average cross section. A novel formula for average cross section in any dimension is presented. The equation of the stationary shape is derived and the general characteristics are discussed through algebraic solution, numerical analysis, and interpretation with graphical display. The special properties of the stationary shapes in 2, 3, and 5 dimension is examined.