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OP-7435Status unassessedAnalysis

Log-Brunn-Minkowski Inequality

Symmetry and convexity in geometric inequalities · Historical workshop question

MSC 46 · Functional analysis

Editorial mapping of AIMPL source category: Functional analysis. http://aimpl.org/symconvgeomineq/ Not an author-supplied MSC code.

Historical list · status unassessedAs of 2026-09-08Source & evidence ↓
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Problem statement

Show that if KK and LL symmetric convex bodies in Rn\mathbb{R}^n, then for all 0≤t≤10 \leq t \leq 1 the following hods

∣tK+∘(1−t)L∣≥∣K∣t∣L∣1−t|tK +_\circ (1-t)L| \geq |K|^t |L|^{1-t}

with equality if and only if K=LK = L or KK and LL are parallelograms with parallel sides.

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Background

Question context

For convex bodies KK and LL in Rn\mathbb{R}^n with support functions hKh_K and hLh_L the logarithmic sum tK+∘(1−t)LtK +_\circ (1-t)L is defined as

tK+∘(1−t)L=∩u∈Sn−1{x∈Rn: ⟨x,u⟩≤hK(u)thL(u)1−t}tK +_\circ (1-t)L = \underset{u \in S^{n-1}}{\cap} \left\{x \in \mathbb{R}^n:\ \langle x, u\rangle \leq h_K(u)^th_L(u)^{1-t} \right\}

where hK(u)=max⁡x∈K⟨x,u⟩.h_K(u) = \max\limits_{x \in K} \langle x, u\rangle .

Progress & source assessment

Historical workshop record. Current open/solved status has not been independently reassessed.

Status note in the source

The possible way to approach the problem is to find some functional versions for logarithmic Brunn-Minkowski inequality.

References
  1. [1]Symmetry and convexity in geometric inequalities link

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