# Affine quermassintegrals of random polytopes

## TL;DR

This paper investigates affine quermassintegrals of convex bodies, providing affirmative results for certain random polytopes and establishing bounds for specific bodies like the $ ext{l}_1^n$ ball, with implications for unconditional convex bodies.

## Contribution

It offers new bounds for affine quermassintegrals of random polytopes and explores their behavior for specific convex bodies, advancing understanding of Lutwak's conjectures.

## Key findings

- Affirmative bounds for random polytopes' affine quermassintegrals.
- Upper bounds for $ ext{l}_1^n$ ball's affine quermassintegrals.
- Implications for unconditional convex bodies.

## Abstract

A question related to some conjectures of Lutwak about the affine quermassintegrals of a convex body $K$ in ${\mathbb R}^n$ asks whether for every convex body $K$ in ${\mathbb R}^n$ and all $1\leqslant k\leqslant n$ $$\Phi_{[k]}(K):={\rm vol}_n(K)^{-\frac{1}{n}}\left (\int_{G_{n,k}}{\rm vol}_k(P_F(K))^{-n}\,d\nu_{n,k}(F)\right )^{-\frac{1}{kn}}\leqslant c\sqrt{n/k},$$ where $c>0$ is an absolute constant. We provide an affirmative answer for some broad classes of random polytopes. We also discuss upper bounds for $\Phi_{[k]}(K)$ when $K=B_1^n$, the unit ball of $\ell_1^n$, and explain how this special instance has implications for the case of a general unconditional convex body $K$.