# Nonnegative forms with sublevel sets of minimal volume

arXiv: https://arxiv.org/abs/1909.10935 · HTML: https://mlsanity.com/paper/1909.10935/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Sep 2019.

## Built on by (most cited first, 1 of 1)

- [Offline Guarded Safe Reinforcement Learning for Medical Treatment Optimization Strategies](https://mlsanity.com/paper/2505.16242/) (2025)

## Abstract (arXiv)

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.