  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).
