# Chebyshev centers and radii for sets induced by quadratic matrix inequalities

## TL;DR

This paper provides mathematical formulas for calculating key geometric properties of matrix sets defined by quadratic inequalities.

## Contribution

The paper derives closed-form solutions for Chebyshev centers, radii, and diameters of sets induced by quadratic matrix inequalities.

## Key findings

- Closed-form solutions are given for Chebyshev centers and radii under various unitarily invariant norms.
- The diameter of the set is also computed in closed form.
- Applications in data-driven modeling and control are discussed.

## Abstract

This paper studies sets of matrices induced by quadratic inequalities. In particular, the center and radius of a smallest ball containing the set, called a Chebyshev center and the Chebyshev radius, are studied. In addition, this work studies the diameter of the set, which is the farthest distance between any two elements of the set. Closed-form solutions are provided for a Chebyshev center, the Chebyshev radius, and the diameter of sets induced by quadratic matrix inequalities (QMIs) with respect to arbitrary unitarily invariant norms. Examples of these norms include the Frobenius norm, spectral norm, nuclear norm, Schatten p-norms, and Ky Fan k-norms. In addition, closed-form solutions are presented for the radius of the largest ball within a QMI-induced set. Finally, the paper discusses applications of the presented results in data-driven modeling and control.