Geometry of generalized virtual polyhedra

TL;DR

This paper explores the connections between generalized virtual polyhedra, torus manifolds, and classical convex body theory, highlighting their differences and relationships without providing detailed proofs.

Contribution

It clarifies how recent generalizations of virtual polyhedra relate to classical convex geometry and homotopy theory, bridging different mathematical frameworks.

Findings

Relates virtual polyhedra to torus manifolds and convex bodies.

Highlights differences between homotopy-based and Euler characteristic-based approaches.

Provides a conceptual overview without detailed proofs.

Abstract

Partial generalizations of virtual polyhedra theory (sometimes under different names) appeared recently in the theory of torus manifolds. These generalizations look very different from the original virtual polyhedra theory. They are based on simple arguments from homotopy theory while the original theory is based on integration over Euler characteristic. In the paper we explain how these generalizations are related to the classical theory of convex bodies and to the original virtual polyhedra theory. The paper basically contains no proofs: all proofs and all details can be found in the cited literature. The paper is based on my talk dedicated to V. I. Arnold's 85-th anniversary at the International Conference on Differential Equations and Dynamical Systems 2022 (Suzdal).