Inequalities for the surface area of projections of convex bodies

TL;DR

This paper establishes inequalities relating the surface area of convex bodies to the surface areas of their projections, extending classical geometric inequalities and analyzing the dependence on dimension and position.

Contribution

It introduces new inequalities comparing surface area of convex bodies to that of their projections, including quermassintegrals, with analysis of dimension dependence and special positions.

Findings

Derived inequalities for surface area and projections of convex bodies.

Analyzed the dependence of constants on dimension and position.

Extended classical geometric inequalities to surface area and projections.

Abstract

We provide general inequalities that compare the surface area S(K) of a convex body K in ${\mathbb R}^n$ to the minimal, average or maximal surface area of its hyperplane or lower dimensional projections. We discuss the same questions for all the quermassintegrals of K. We examine separately the dependence of the constants on the dimension in the case where K is in some of the classical positions or K is a projection body. Our results are in the spirit of the hyperplane problem, with sections replaced by projections and volume by surface area.