# Bi-log-concavity: some properties and some remarks towards a multi-dimensional extension

## TL;DR

This paper explores properties of bi-log-concave probability measures, extending key features of log-concavity to a broader class, and discusses potential multi-dimensional generalizations.

## Contribution

It computes the isoperimetric constant for bi-log-concave measures, extends properties like tail decay and convolution stability, and proposes a framework for multi-dimensional extension.

## Key findings

- Bi-log-concave measures have exponentially decreasing tails.
- Convolution of bi-log-concave and log-concave measures remains bi-log-concave.
- Smooth densities are dense in bi-log-concave densities under $L_p$ norms.

## Abstract

Bi-log-concavity of probability measures is a univariate extension of the notion of log-concavity that has been recently proposed in a statistical literature. Among other things, it has the nice property from a modelisation perspective to admit some multimodal distributions, while preserving some nice features of log-concave measures. We compute the isoperimetric constant for a bi-log-concave measure, extending a property available for log-concave measures. This implies that bi-log-concave measures have exponentially decreasing tails. Then we show that the convolution of a bi-log-concave measure with a log-concave one is bi-log-concave. Consequently, infinitely differentiable, positive densities are dense in the set of bi-log-concave densities for $L_p-$norms, $p \in [1;+\infty]$. We also derive a necessary and sufficient condition for the convolution of two bi-log-concave measures to be bi-log-concave. We conclude this note by discussing ways of defining a multi-dimensional extension of the notion of bi-log-concavity. We propose an approach based on a variant of the isoperimetric problem, restricted to half-spaces.