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.

中文翻译：

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具有最小体积的子级集的非负形式

我们证明欧几里得球在有界 Bombieri 范数的非负形式的子级集以及其 Gram 矩阵有界 Frobenius 或核（或更一般地说，p-Schatten）范数的平方和形式的子级集中具有最小的体积. 欧几里得球的这些体积最小化特性就其表示（作为一种固定偶数形式的子级集）补充了其众多固有的几何特性。我们还提供了对结果的概率解释。