# Lutwak-Petty projection inequalities for Minkowski valuations and their duals

## TL;DR

This paper extends Lutwak's volume inequalities to Minkowski valuations and their duals, introducing new integral representations and generalizations that connect to affine quermassintegrals and intersection bodies.

## Contribution

It generalizes Lutwak's inequalities to a broader class of Minkowski valuations and their duals, using novel integral representations and establishing new relations with affine quermassintegrals.

## Key findings

- Lutwak's inequalities are the strongest among the generalized families.
- A new integral representation isolates Lutwak's inequalities.
- Dual inequalities for intersection bodies are established.

## Abstract

Lutwak's volume inequalities for polar projection bodies of all orders are generalized to polarizations of Minkowski valuations generated by even, zonal measures on the Euclidean unit sphere. This is based on analogues of mixed projection bodies for such Minkowski valuations and a generalization of the notion of centroid bodies. A new integral representation is used to single out Lutwak's inequalities as the strongest among these families of inequalities, which in turn are related to a conjecture on affine quermassintegrals. In the dual setting, a generalization of volume inequalities for intersection bodies of all orders by Leng and Lu is proved. These results are related to Grinberg's inequalities for dual affine quermassintegrals.