Equality in Liakopoulos's generalized dual Loomis-Whitney inequality via Barthe's Reverse Brascamp-Lieb inequality

TL;DR

The paper characterizes when Liakopoulos's generalized dual Loomis–Whitney inequality becomes an equality by leveraging Barthe's Geometric Reverse Brascamp–Lieb inequality. It shows that equality forces a direct-sum decomposition of the ambient space into independent subspaces and that the convex body is the convex hull of its sections along those subspaces. The approach relies on the structure of extremizers from the reverse Brascamp–Lieb framework and log-concavity properties, tying the equality case to a rigid geometric decomposition. This yields a precise, verifiable condition with potential implications for stability analyses of Brascamp–Lieb-type inequalities.

Abstract

We use the characterization of the case of equality in Barthe's Geometric Reverse Brascamp-Lieb inequality to characterize equality in Liakopoulos's volume estimate in terms of sections by certain lower-dimensional linear subspaces.

Theorems & Definitions (14)

• Theorem 1: Loomis, Whitney
• Theorem 2: Bollobás, Thomason
• Theorem 3: Folklore
• Theorem 4: Meyer
• Theorem 5: Liakopoulos
• Theorem 6
• Theorem 7: Brascamp, Lieb, Ball, Barthe
• Remark 8
• Theorem 9: Valdimarsson
• Theorem 10: Barthe