Godbersen's conjecture for locally anti-blocking bodies

TL;DR

This work proves Godbersen's conjecture for locally anti-blocking bodies by establishing $V_n(K[j],-K[n-j])\le \binom{n}{j}\mathrm{Vol}(K)$ for $0<j<n$, with equality forcing $K$ to be a simplex. The authors reduce the problem to an orthant decomposition $K=\bigcup_\sigma K_\sigma$, apply LAB-mixvol identities and Rogers–Shephard-type inequalities, and use a key lemma on mixed volumes of coordinate-aligned simplices to analyze equality cases. They show all equality cases arise only when each orthant piece is a simplex and, after a dimension-reduction argument, that the global extremal shapes are simplices up to linear changes. Consequently, simplices are the sole equality extremals among locally anti-blocking bodies, strengthening evidence for Godbersen's conjecture in this natural body class.

Abstract

In this note we give a short proof of Godbersen's conjecture for the class of locally anti-blocking bodies. We show that all equality cases amongst locally anti-blocking bodies are for simplices, further supporting the conjecture. The proof of equality cases introduces a useful calculation of mixed volumes of aligned simplices.

Theorems & Definitions (16)

• Conjecture 1
• Theorem 1.1
• Lemma 1.2
• Lemma 2.1
• Lemma 2.2
• proof
• proof
• Theorem 2.4
• proof : Proof of the inequality in Theorem \ref{['thm:god-for-LAB']}
• proof : Proof of the equality cases in Theorem \ref{['thm:god-for-LAB']}