On a $j$-Santal\'{o} Conjecture

TL;DR

This paper introduces a family of conjectured sharp Santaló inequalities for multiple sets or functions, generalizing classical results and confirming some cases, with extremals related to $l_j^n$-balls and connections to other geometric conjectures.

Contribution

It proposes a new family of $j$-Santaló conjectures, confirms some cases including the extremal cases, and links these inequalities to classical and modern geometric conjectures.

Findings

Confirmed the conjecture for the case j=k and the unconditional case.

Identified extremals as tuples of $l_j^n$-balls.

Strengthened previous results related to the case j=2.

Abstract

Let $k\geq 2$ be an integer. In the spirit of Kolesnikov-Werner \cite{KW}, for each $j\in\{2,\ldots,k\}$, we conjecture a sharp Santal\'{o} type inequality (we call it $j$-Santal\'{o} conjecture) for many sets (or more generally for many functions), which we are able to confirm in some cases, including the case $j=k$ and the unconditional case. Interestingly, the extremals of this family of inequalities are tuples of the $l_j^n$-ball. Our results also strengthen one of the main results in \cite{KW}, which corresponds to the case $j=2$. All members of the family of our conjectured inequalities can be interpreted as generalizations of the classical Blaschke-Santal\'{o} inequality. Related, we discuss an analogue of a conjecture due to K. Ball \cite{Ball-conjecture} in the multi-entry setting and establish a connection to the $j$-Santal\'{o} conjecture.