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Convex bodies in Cn\mathbb{C}^n

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

Consider a convex body in Cn\mathbb{C}^n and fix a real subspace EE of Cn\mathbb{C}^n. Define VE(K)=∫U(n)∣K∣ϕ(E)∣ dϕV_E(K) = \int\limits_{U(n)} |K|\phi(E)| \, d\phi, where U(n)U(n) is a unitary group. Does Brunn-Minkowski type inequality hold for such quantities:

VE(K+L)1i≥VE(K)1i+VE(L)1i?V_E(K + L)^{\frac1i} \geq V_E(K)^{\frac1i} +V_E(L)^{\frac1i}?

Also, for what type of subspaces EE this will work?

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

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