# Inequalities between $L^p$-norms for log-concave distributions

## TL;DR

This paper establishes inequalities between Lp-norms for log-concave distributions, generalizing bounds on differential entropy and expanding classical Lp-norm inequalities within Euclidean space.

## Contribution

It introduces new inequalities between Lp-norms specifically for log-concave distributions, extending existing bounds on differential entropy.

## Key findings

- Inequalities generalize bounds on differential entropy.
- Results apply to important distributions like normal and exponential.
- Provides a broader understanding of Lp-norm relationships for log-concave measures.

## Abstract

Log-concave distributions include some important distributions such as normal distribution, exponential distribution and so on. In this note, we show inequalities between two Lp-norms for log-concave distributions on the Euclidean space. These inequalities are the generalizations of the upper and lower bound of the differential entropy and are also interpreted as a kind of expansion of the inequality between two Lp-norms on the measurable set with finite measure.