Kaehler Geometry, Momentum Maps and Convex Sets

TL;DR

This paper explores the connections between Kähler geometry, momentum maps, and convex sets, providing detailed insights into Gromov's approach to inequalities in convex and Kähler geometry.

Contribution

It offers a detailed exposition of Gromov's proof techniques linking convex geometry and Kähler manifolds, emphasizing the mathematical methods involved.

Findings

Gromov's proof of Alexandrov--Fenchel inequalities for convex sets.

Extension of Brunn--Minkowski inequality to Kähler manifolds.

Detailed analysis of techniques from various mathematical areas.

Abstract

These notes grew out of an expose on M. Gromov's paper "Convex sets and K\"ahler manifolds'' ("Advances in Differential Geometry and Topology,'' World Scientific, 1990) at the DMV-Seminar on "Combinatorical Convex Geometry and Toric Varieties'' in Blaubeuren in April `93. Gromov's paper deals with a proof of Alexandrov--Fenchel type inequalities and the Brunn--Minkowski inequality for finite dimensional compact convex sets and their variants for compact K\"ahler manifolds. The emphasis of these notes lies on basic details and the techniques from various mathematical areas involved in Gromov's arguments.