Exact asymptotic volume and volume ratio of Schatten unit balls

@article{Kabluchko2020ExactAV,
  title={Exact asymptotic volume and volume ratio of Schatten unit balls},
  author={Zakhar Kabluchko and Joscha Prochno and Christoph Thaele},
  journal={J. Approx. Theory},
  year={2020},
  volume={257},
  pages={105457}
}
The unit ball $B_p^n(\mathbb{R})$ of the finite-dimensional Schatten trace class $\mathcal S_p^n$ consists of all real $n\times n$ matrices $A$ whose singular values $s_1(A),\ldots,s_n(A)$ satisfy $s_1^p(A)+\ldots+s_n^p(A)\leq 1$, where $p>0$. Saint Raymond [Studia Math.\ 80, 63--75, 1984] showed that the limit $$ \lim_{n\to\infty} n^{1/2 + 1/p} \big(\text{Vol}\, B_p^n(\mathbb{R})\big)^{1/n^2} $$ exists in $(0,\infty)$ and provided both lower and upper bounds. In this paper we determine the… Expand

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