Polytopal balls arising in optimization

TL;DR

This paper investigates a family of polytopes and their duals that serve as unit balls in various optimization problems, analyzing their combinatorial and geometric properties.

Contribution

It introduces and studies a new family of polytopes interpolating between hypercubes and cross-polytopes, providing detailed combinatorial and geometric insights.

Findings

Determined the f-vector of the polytopes.

Computed volumes of the polytopes and their boundaries.

Established geometric properties related to optimization applications.

Abstract

We study a family of polytopes and their duals, that appear in various optimization problems as the unit balls for certain norms. These two families interpolate between the hypercube, the unit ball for the $\infty$-norm, and its dual cross-polytope, the unit ball for the $1$-norm. We give combinatorial and geometric properties of both families of polytopes such as their $f$-vector, their volume, and the volume of their boundary.