A Simple Proof of Cauchy's Surface Area Formula

TL;DR

This paper provides a concise and straightforward proof of Cauchy's surface area formula, establishing a fundamental relationship between the average projection area of convex bodies and their surface area.

Contribution

It introduces a simplified proof of Cauchy's surface area formula, making the understanding of this geometric relationship more accessible.

Findings

Proof is shorter and simpler than previous versions

Confirms the proportional relationship between projection averages and surface area

Enhances theoretical understanding of convex geometry

Abstract

We give a short and simple proof of Cauchy's surface area formula, which states that the average area of a projection of a convex body is equal to its surface area up to a multiplicative constant in the dimension.