Planar Projections and Second Intrinsic Volume

TL;DR

This paper investigates the dependence of geometric measures like area and perimeter in random projections of polyhedra, revealing positive correlation for cubes and negative for tetrahedra, through generalizations of mean width.

Contribution

It introduces new insights into the dependence structures of geometric measures in random projections using generalized mean width concepts.

Findings

Area and perimeter of a cube are positively correlated (correlation 0.915).

Area and perimeter of a tetrahedron appear negatively dependent.

Generalizations of mean width are used to analyze these dependencies.

Abstract

Consider random shadows of a cube and of a regular tetrahedron. Area and perimeter of the former are positively dependent (with correlation 0.915...), whereas area and perimeter of the latter appear to be negatively dependent. This is only one result of many, all involving generalizations of mean width.