The Data Singular and the Data Isotropic Loci for Affine Cones

TL;DR

This paper explores the geometric properties of special loci related to the Euclidean distance degree of affine cones, focusing on their connections to dual cones and the impact on critical point counts.

Contribution

It introduces the concepts of Euclidean Distance Data Singular and Isotropic Loci for affine cones and analyzes their relationships with dual cones.

Findings

Identifies the geometric nature of the singular and isotropic loci.

Establishes connections between these loci and the dual cone structure.

Provides insights into how these loci influence the critical points of the Euclidean distance function.

Abstract

The generic number of critical points of the Euclidean distance function from a data point to a variety is called the Euclidean distance degree. The two special loci of the data points where the number of critical points is smaller then the ED degree are called the Euclidean Distance Data Singular Locus and the Euclidean Distance Data Isotropic Locus. In this article we present connections between these two special loci of an affine cone and its dual cone.