{"abstracts":[{"sha1":"3c99b2c84d271d31687dd776f088539aab8f6b23","content":"The unit ball $B_p^n(\\mathbb{R})$ of the finite-dimensional Schatten trace\nclass $\\mathcal S_p^n$ consists of all real $n\\times n$ matrices $A$ whose\nsingular values $s_1(A),\\ldots,s_n(A)$ satisfy $s_1^p(A)+\\ldots+s_n^p(A)\\leq\n1$, where $p>0$. Saint Raymond [Studia Math.\\ 80, 63--75, 1984] showed that the\nlimit $$ \\lim_{n\\to\\infty} n^{1/2 + 1/p} \\big(\\text{Vol}\\,\nB_p^n(\\mathbb{R})\\big)^{1/n^2} $$ exists in $(0,\\infty)$ and provided both\nlower and upper bounds. In this paper we determine the precise limiting\nconstant based on ideas from the theory of logarithmic potentials with external\nfields. A similar result is obtained for complex Schatten balls. As an\napplication we compute the precise asymptotic volume ratio of the Schatten\n$p$-balls, as $n\\to\\infty$, thereby extending Saint Raymond's estimate in the\ncase of the nuclear norm ($p=1$) to the full regime $1\\leq p \\leq \\infty$ with\nexact limiting behavior.","mimetype":"application/x-latex","lang":"en"},{"sha1":"44b40c51d1a1a0cfee5c3fa1a79c72f47618087d","content":"The unit ball B_p^n(R) of the finite-dimensional Schatten trace\nclass S_p^n consists of all real n× n matrices A whose\nsingular values s_1(A),...,s_n(A) satisfy s_1^p(A)+...+s_n^p(A)≤\n1, where p>0. Saint Raymond [Studia Math. 80, 63--75, 1984] showed that the\nlimit _n→∞ n^1/2 + 1/p(Vol \nB_p^n(R))^1/n^2 exists in (0,∞) and provided both\nlower and upper bounds. In this paper we determine the precise limiting\nconstant based on ideas from the theory of logarithmic potentials with external\nfields. A similar result is obtained for complex Schatten balls. As an\napplication we compute the precise asymptotic volume ratio of the Schatten\np-balls, as n→∞, thereby extending Saint Raymond's estimate in the\ncase of the nuclear norm (p=1) to the full regime 1≤ p ≤∞ with\nexact limiting behavior.","mimetype":"text/plain","lang":"en"}],"refs":[],"contribs":[{"index":0,"raw_name":"Zakhar Kabluchko","role":"author"},{"index":1,"raw_name":"Joscha Prochno","role":"author"},{"index":2,"raw_name":"Christoph Thaele","role":"author"}],"license_slug":"ARXIV-1.0","language":"en","version":"v1","ext_ids":{"arxiv":"1804.03467v1"},"release_year":2018,"release_date":"2018-04-10","release_stage":"submitted","release_type":"article","webcaptures":[],"filesets":[],"files":[{"release_ids":["7ncecl64mrecrlzgeiyqb7vl4e"],"mimetype":"application/pdf","urls":[{"url":"https://web.archive.org/web/20200827095748/https://arxiv.org/pdf/1804.03467v1.pdf","rel":"webarchive"},{"url":"https://arxiv.org/pdf/1804.03467v1.pdf","rel":"repository"}],"sha256":"02402056a6404f095fe135e4ab82134396403efd1c239be8bbfd660e00465d05","sha1":"3e6c2def557e8925e9bb94d4b5712f6d69bbdddc","md5":"69347785e9d9f00253d0a7bad70a210c","size":206007,"revision":"1bb23542-74ad-4288-aba7-64eaf5963b78","ident":"ntmv6lklurbmjlagwiqylb5x44","state":"active"}],"work_id":"cewzuai5pfa5livtqxhji4tufy","title":"Exact asymptotic volume and volume ratio of Schatten unit balls","state":"active","ident":"7ncecl64mrecrlzgeiyqb7vl4e","revision":"925546ab-d09e-4b26-97e0-297efe28badc","extra":{"arxiv":{"base_id":"1804.03467","categories":["math.FA","math.MG","math.PR"]}}}