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bibkey: faifmanpolterovich2026faberkrahn
authors: D. Faifman; I. Polterovich
year: 2026
title: "The Faber–Krahn position of convex bodies and Gaussian measure inequalities"
doi: 10.48550/arXiv.2607.21539
url: https://arxiv.org/abs/2607.21539v2
claim: "Question 5.3. Let K ⊂ R^n be a convex body whose Lebesgue barycenter is at the origin. Is t ↦ γ_n(e^t K) log-concave? More generally, is t ↦ γ_n(e^{tS} K) log-concave for every symmetric matrix S?"
strata_touched:
  - D5/S3/Geometry/CentredGaussianDilationRefutation
license: citation-only
triage: anchor

Locator

Author text: https://arxiv.org/abs/2607.21539v2.

• DOI: https://doi.org/10.48550/arXiv.2607.21539

D. Faifman and I. Polterovich, The Faber–Krahn position of convex bodies and Gaussian measure inequalities, arXiv:2607.21539v2, §5.2, p. 38, the paragraph preceding Question 5.3 and Question 5.3 itself.

Centred Gaussian dilation

The passage states verbatim:

The B-theorem requires central symmetry. To prove Theorem 1.1, we circumvent this by proving a weaker concavity property of the Gaussian measure in a special case. However, we do not know a convex counterexample to the B-conjecture with its linear barycenter at the origin. Question 5.3. Let K ⊂ R^n be a convex body whose Lebesgue barycenter is at the origin. Is t ↦ γ_n(e^t K) log-concave? More generally, is t ↦ γ_n(e^{tS} K) log-concave for every symmetric matrix S?

Here ${\gamma }_n$ is standard Gaussian measure, the barycenter uses Lebesgue measure, and the dilations are about the origin. The scalar case corresponds to the identity matrix in the more general clause.

ASSUMED-UNVERIFIED

The locator and quotation are supplied with the statement. The author text has not been independently retrieved; an exhaustive literature or priority claim is unverified.