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Symmetry and convexity · Question 3

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

Let p>1p > 1 and H⊂RnH \subset \mathbb{R}^n of dimension kk. Consider a map

f(t1,…,tn)=∣(et10⋯00et2⋯0⋮⋮⋱⋮00⋯etn)Bpn∩H∣k,f(t_1, \dots, t_n) = \left| \begin{pmatrix} e^{t_1} & 0 & \cdots & 0 \\ 0 & e^{t_2} & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & e^{t_n} \end{pmatrix} B_p^n \cap H \right|_k,

where Bpn={x∈Rn: ∣x1∣p+⋯+∣xn∣pp≤1}B_p^n = \{ x\in \mathbb{R}^n: \ \sqrt[p]{|x_1|^p + \cdots + |x_n|^p} \leq 1\}.

Is ff log-concave jointly in t1,…,tnt_1, \dots, t_n?

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

Status note in the source

For cases p=1,2p = 1, 2, the answer is given by Tkocz, Nayar. Note that when p=∞p = \infty, the statement is equivalent to log-Brunn-Minkowski.

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

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