Nonnegative forms with sublevel sets of minimal volume

TL;DR

This paper proves that the Euclidean ball has the smallest volume among certain nonnegative polynomial sublevel sets with bounded norms, highlighting its optimal geometric properties and providing a probabilistic interpretation.

Contribution

It establishes the minimal volume property of the Euclidean ball among sublevel sets of nonnegative forms with bounded Bombieri, Frobenius, nuclear, or p-Schatten norms, extending geometric understanding.

Findings

Euclidean ball minimizes volume among sublevel sets of nonnegative forms with bounded norms

Results apply to sum of squares forms with Gram matrices of bounded Frobenius or nuclear norm

Probabilistic interpretation of volume-minimizing properties provided

Abstract

We show that the Euclidean ball has the smallest volume among sublevel sets of nonnegative forms of bounded Bombieri norm as well as among sublevel sets of sum of squares forms whose Gram matrix has bounded Frobenius or nuclear (or, more generally, p-Schatten) norm. These volume-minimizing properties of the Euclidean ball with respect to its representation (as a sublevel set of a form of fixed even degree) complement its numerous intrinsic geometric properties. We also provide a probabilistic interpretation of the results.