Two new proofs of partial Godbersen's Conjecture

TL;DR

This paper presents two novel proofs of Godbersen's conjecture, one using Helly's theorem and the other employing the Brunn-Minkowski inequality, offering new perspectives and methods in convex geometry.

Contribution

It introduces two new proofs of Godbersen's conjecture, one concise and elegant via Helly's theorem, and another based on the Brunn-Minkowski inequality, expanding the toolkit for convex geometric inequalities.

Findings

Proof using Helly's theorem simplifies the inequality proof.

Brunn-Minkowski-based proof establishes the inclusion $-K extless nK$.

Both proofs confirm Godbersen's conjecture.

Abstract

Two new proofs are provided, offering two new perspectives on Godbersen's conjecture. One of the proofs utilizes Helly's theorem to provide a concise and elegant proof of the inequality in Godbersen's conjecture. The other proof utilizes the Brunn-Minkowski inequality to provide a completely new proof of the inclusion $-K\subset nK$ for convex bodies $K$ with centroid at the origin, thereby proving Godbersen's conjecture.