A remark on the slicing problem

TL;DR

This paper links the slicing problem to a specific integral parameter involving convex bodies and establishes bounds that relate to the isotropic constant, advancing understanding of high-dimensional convex geometry.

Contribution

It introduces a reduction of the slicing problem to a study of a particular integral parameter and derives bounds connecting this parameter to the isotropic constant.

Findings

Reduction of the slicing problem to the study of I_1(K,Z_q^o(K))

Derived bounds on the isotropic constant L_n based on the parameter I_1

Established a relationship between the parameter bounds and the dimension n

Abstract

The purpose of this article is to describe a reduction of the slicing problem to the study of the parameter I_1(K,Z_q^o(K))=\int_K ||< :, x> ||_{L_q(K)}dx. We show that an upper bound of the form I_1(K,Z_q^o(K))\leq C_1q^s\sqrt{n}L_K^2, with 1/2\leq s\leq 1, leads to the estimate L_n\leq \frac{C_2\sqrt[4]{n}log(n)} {q^{(1-s)/2}}, where L_n:= max {L_K : K is an isotropic convex body in R^n}.