How do difference bodies in complex vector spaces look like? A geometrical approach

TL;DR

This paper explores the geometric properties of the complex difference body in complex vector spaces, establishing new inequalities and characterizations, and highlighting differences from the classical real case.

Contribution

It provides an equivalent support function expression, characterizes when the complex difference body is a ball or a polytope, and relates its properties to those of the original bodies.

Findings

The dimension of the complex difference body depends on its position relative to the complex structure.

Bodies for which the complex difference body is a ball are characterized using spherical harmonics.

The complex difference body is a polytope if and only if the original bodies are polytopes.

Abstract

We investigate geometrical properties and inequalities satisfied by the complex difference body, in the sense of studying which of the classical ones for the difference body have an analog in the complex framework. Among others we give an equivalent expression for the support function of the complex difference body and prove that, unlike the classical case, the dimension of the complex difference body depends on the position of the body with respect to the complex structure of the vector space. We use spherical harmonics to characterize the bodies for which the complex difference body is a ball, we prove that it is a polytope if and only if the two bodies involved in the construction are polytopes and provide several inequalities for classical magnitudes of the complex difference body, as volume, quermassintegrals and diameter, in terms of the corresponding ones for the involved bodies.