# Affine isoperimetric inequalities on flag manifolds

## TL;DR

This paper introduces new affine quermassintegrals on flag manifolds, generalizing classical geometric measures, and establishes related affine isoperimetric inequalities and their functional versions, extending convex geometry results.

## Contribution

It defines ${f r}$-flag affine quermassintegrals and their duals, extending affine geometric measures to flag manifolds with invariance properties and new inequalities.

## Key findings

- Established affine and linear invariance properties.
- Proved affine isoperimetric inequalities and their approximate reverses.
- Introduced functional forms and corresponding inequalities.

## Abstract

Building on work of Furstenberg and Tzkoni, we introduce ${\bf r}$-flag affine quermassintegrals and their dual versions. These quantities generalize affine and dual affine quermassintegrals as averages on flag manifolds (where the Grassmannian can be considered as a special case). We establish affine and linear invariance properties and extend fundamental results to this new setting. In particular, we prove several affine isoperimetric inequalities from convex geometry and their approximate reverse forms. We also introduce functional forms of these quantities and establish corresponding inequalities.