$L_p$ John ellipsoids for log-concave functions

TL;DR

This paper extends the concept of $L_p$ John ellipsoids to log-concave functions, characterizing their properties and establishing related inequalities, including an analog of Ball's volume ratio inequality.

Contribution

It introduces the $L_p$ John ellipsoid for log-concave functions, building on $L_p$ Minkowski theory, and proves new inequalities including a volume ratio inequality.

Findings

Characterization of $L_p$ John ellipsoid for log-concave functions

Establishment of inequalities related to $L_p$ John ellipsoid

Proof of an analog of Ball's volume ratio inequality

Abstract

The aim of this paper is to develop the $L_p$ John ellipsoid for the geometry of log-concave functions. Using the results of the $L_p$ Minkowski theory for log-concave function established in \cite{fan-xin-ye-geo2020}, we characterize the $L_p$ John ellipsoid for log-concave function, and establish some inequalities of the $L_p$ John ellipsoid for log-concave function. Finally, the analog of Ball's volume ratio inequality for the $L_p$ John ellipsoid of log-concave function is established.