Topology of the Bend Loci of Convex Piecewise Linear Functions

TL;DR

This paper investigates the topological properties of the bend loci of convex piecewise linear functions, providing new insights into their structure and connectivity.

Contribution

It establishes the connectivity of intersections of generic polyhedral hypersurfaces, extending understanding of their topological complexity.

Findings

Complete intersections are (d-n-1)-connected for d ≥ 2 and d > n.

Provides a proof for the connectivity of these intersections.

Enhances theoretical understanding of polyhedral hypersurface topology.

Abstract

This short article serves as the appendix for [Tran and Wang, 2022]. We prove that a complete intersection of $n$ generic polyhedral hypersurfaces in $\mathbb{R}^d$ is $(d-n-1)$-connected for $d\geq 2, d>n$.