{"paper":{"title":"Exact asymptotic volume and volume ratio of Schatten unit balls","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG","math.PR"],"primary_cat":"math.FA","authors_text":"Christoph Thaele, Joscha Prochno, Zakhar Kabluchko","submitted_at":"2018-04-10T11:58:51Z","abstract_excerpt":"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 precise limiting constant based on ideas from the theory of logarithmic potentials with external fields."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.03467","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}