Quantitative anisotropic isoperimetric and Brunn−Minkowski inequalities for convex sets with improved defect estimates

Abstract

In this paper we revisit the anisotropic isoperimetric and the Brunn-Minkowski inequalities for convex sets. The best known constant C(n) = Cn 7 depending on the space dimension n in both inequalities is due to Segal [35]. We improve that constant to Cn 6 for convex sets and to Cn 5 for centrally symmetric convex sets. We also conjecture, that the best constant in both inequalities must be of the form Cn 2 , i.e., quadratic in n. The tools are the Brenier's mapping from the theory of mass transportation combined with new sharp geometric-arithmetic mean and some algebraic inequalities plus a trace estimate by Figalli, Maggi and Pratelli.

The logarithmic Minkowski conjecture and the Lp-Minkowski problem

“…The current best known bound for γ * (n) is cn −5 (log n) −10 , which follows by combining the general estimate of Kolesnikov-Milman [169], Theorem 12.2, with the polylogarithmic bound of Klartag, Lehec [164] on the Cheeger constant of a convex body in isotropic position improving on Yuansi Chen's work [76] on the Kannan-Lovász-Simonovits conjecture. Harutyunyan [137] conjectured that γ * (n) = cn −2 is the optimal order of the constant, and showed that it can't be of smaller order. Actually, Segal [232] observed that Dar's conjecture in [88] would imply that we may choose γ * (n) = cn −2 for some absolute constant c > 0.…”

The Logarithmic Minkowski conjecture and the $L_p$-Minkowski Problem

“…The current best known bound for γ * (n) is cn −5 (log n) −10 , which follows by combining the general estimate of Kolesnikov-Milman [158], Theorem 12.2, with the polylogarithmic bound of Klartag, Lehec [153] on the Cheeger constant of a convex body in isotropic position improving on Yuansi Chen's work [73] on the Kannan-Lovász-Simonovits conjecture. Harutyunyan [128] conjectured that γ * (n) = cn −2 is the optimal order of the constant, and showed that it can't be of smaller order. Actually, Segal [215] observed that Dar's conjecture in [82] would imply that we may choose γ * (n) = cn −2 for some absolute constant c > 0.…”