# 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}
}
• Published 10 April 2018
• Computer Science, Mathematics
• J. Approx. Theory
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
9 Citations

#### Topics from this paper

Sanov-type large deviations in Schatten classes
• Mathematics
• 2018
Denote by $\lambda_1(A), \ldots, \lambda_n(A)$ the eigenvalues of an $(n\times n)$-matrix $A$. Let $Z_n$ be an $(n\times n)$-matrix chosen uniformly at random from the matrix analogue to theExpand
Asymptotic estimates for the largest volume ratio of a convex body
• Mathematics
• Revista Matemática Iberoamericana
• 2019
The largest volume ratio of given convex body $K \subset \mathbb{R}^n$ is defined as $$\mbox{lvr}(K):= \sup_{L \subset \mathbb{R}^n} \mbox{vr}(K,L),$$ where the $\sup$ runs over all the convex bodiesExpand
Gelfand numbers of embeddings of Schatten classes
• Mathematics
• 2020
Let $0<p,q\leq \infty$ and denote by $\mathcal{S}_p^N$ and $\mathcal{S}_q^N$ the corresponding Schatten classes of real $N\times N$ matrices. We study the Gelfand numbers of natural identitiesExpand
Intersection of unit balls in classical matrix ensembles
• Mathematics
• 2018
We study the volume of the intersection of two unit balls from one of the classical matrix ensembles GOE, GUE and GSE, as the dimension tends to infinity. This can be regarded as a matrix analogue ofExpand
Approximation, Gelfand, and Kolmogorov numbers of Schatten class embeddings
• Computer Science, Mathematics
• ArXiv
• 2021
This work studies approximation quantities of natural identities S N p,→ S N q between Schatten classes and proves asymptotically sharp bounds up to constants only depending on p and q, showing how approximation numbers are intimately related to the Gelfand numbers and their duals, the Kolmogorov numbers. Expand
The maximum entropy principle and volumetric properties of Orlicz balls
• Mathematics
• 2020
Abstract We study the precise asymptotic volume of balls in Orlicz spaces and show that the volume of the intersection of two Orlicz balls undergoes a phase transition when the dimension of theExpand
Weighted $p$-radial Distributions on Euclidean and Matrix $p$-balls with Applications to Large Deviations
• Mathematics
• 2021
Abstract. A probabilistic representation for a class of weighted p-radial distributions, based on mixtures of a weighted cone probability measure and a weighted uniform distribution on the EuclideanExpand
Geometry of $\ell_p^n$-balls: Classical results and recent developments
• Mathematics
• 2018
This survey will appear as a chapter in the forthcoming "Proceedings of the HDP VIII Conference".
Asymptotics of the inertia moments and the variance conjecture in Schatten balls
• Mathematics
• 2021
We study the limit, as the dimension goes to infinity, of the moments of the Hilbert-Schmidt norm of a uniformly distributed matrix in the p-Schatten ball, with entries in the real, complex orExpand

#### References

SHOWING 1-10 OF 43 REFERENCES
Sanov-type large deviations in Schatten classes
• Mathematics
• 2018
Denote by $\lambda_1(A), \ldots, \lambda_n(A)$ the eigenvalues of an $(n\times n)$-matrix $A$. Let $Z_n$ be an $(n\times n)$-matrix chosen uniformly at random from the matrix analogue to theExpand
Entropy numbers of embeddings of Schatten classes
• Mathematics
• 2016
Let $0<p,q \leq \infty$ and denote by $\mathcal S_p^N$ and $\mathcal S_q^N$ the corresponding finite-dimensional Schatten classes. We prove optimal bounds, up to constants only depending on $p$ andExpand
Volume Ratios and a Reverse Isoperimetric Inequality
It is shown that if $C$ is an $n$-dimensional convex body then there is an affine image $\widetilde C$ of $C$ for which $${|\partial \widetilde C|\over |\widetilde C|^{n-1\over n}}$$ is no largerExpand
On the geometry of projective tensor products
• Mathematics
• 2016
In this work, we study the volume ratio of the projective tensor products $\ell^n_p\otimes_{\pi}\ell_q^n\otimes_{\pi}\ell_r^n$ with $1\leq p\leq q \leq r \leq \infty$. We obtain asymptotic formulasExpand
Volume estimates in spaces of homogeneous polynomials
• Mathematics
• 2008
Given m we derive upper and lower estimates for the volume of the unit ball of the Banach space $${\mathcal{P}(^{m}E_{n})}$$ , all m-homogeneous polynomials defined on the nth section En of aExpand
Concentration of mass on the Schatten classes
• Mathematics
• 2007
Abstract Let 1 ⩽ p ⩽ ∞ and B ( S p n ) ˜ be the unit ball of the Schatten trace class of matrices on C n or on R n , normalized to have Lebesgue measure equal to one. We prove that λ ( { T ∈ B ( S pExpand
Intersection of unit balls in classical matrix ensembles
• Mathematics
• 2018
We study the volume of the intersection of two unit balls from one of the classical matrix ensembles GOE, GUE and GSE, as the dimension tends to infinity. This can be regarded as a matrix analogue ofExpand
On the volume of unit balls in Banach spaces
We estimate the volume ratio of lpn ~n lrn, 1 ~ p, r ~ ~, unitary operator ideals and symmetric spaces. We also study the structure of the n-dimensional James pace. We consider the volume of unitExpand
Selberg-type integrals and the variance conjecture for the operator norm
The variance conjecture in Asymptotic Convex Geometry stipulates that the Euclidean norm of a random vector uniformly distributed in a (properly normalised) high-dimensional convex body \$K\subsetExpand
On nearly euclidean decomposition for some classes of Banach spaces
• Mathematics
• 1980
We introduce a new affine invariant of finite dimensional normed space. Using it we show that for some large classes of finite dimensional normed spaces there is a constant C such that every normedExpand