Affine isoperimetric inequalities on flag manifolds
Susanna Dann, Grigoris Paouris, Peter Pivovarov
TL;DR
The paper develops a flag-geometry extension of affine isoperimetric theory by introducing ${\mathbf r}$-flag affine quermassintegrals $\Phi_{\mathbf{r}}(L)$ and their duals $\Psi_{\mathbf{r}}(L)$ on flag manifolds $F^{n}_{\mathbf{r}}$, unifying and extending classical affine quermassintegrals. It proves robust invariance properties under volume-preserving transformations and $SL_n$, derives flag-versions of foundational inequalities (Busemann–Straus–Grinberg, Blaschke–Santaló) with sharp equality cases for ellipsoids, and establishes uniform bounds via Milman’s $M$-ellipsoids. The work further generalizes to permutation-based flag quantities $\Psi_{\omega},\Phi_{\omega}$ with both invariance and counterexamples, and develops functional analogues $I(f)$ and $\Phi_{\mathbf{r}}(f)$ for nonnegative functions, including rearrangement-type bounds. Together, these results expand Brunn–Minkowski theory to flag manifolds and function spaces, offering new tools for convex geometry, geometric analysis, and high-dimensional probability.
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.