{"paper":{"title":"Proportional subspaces of spaces with unconditional basis have good volume properties","license":"","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Marius Junge","submitted_at":"1993-12-29T17:04:17Z","abstract_excerpt":"A generalization of Lozanovskii's result is proved. Let E be $k$-dimensional subspace of an $n$-dimensional Banach space with unconditional basis. Then there exist $x_1,..,x_k \\subset E$ such that $B_E \\p \\subset \\p absconv\\{x_1,..,x_k\\}$ and \\[ \\kla \\frac{{\\rm vol}(absconv\\{x_1,..,x_k\\})}{{\\rm vol}(B_E)} \\mer^{\\frac{1}{k}} \\kl \\kla e\\p \\frac{n}{k} \\mer^2 \\pl .\\] This answers a question of V. Milman which appeared during a GAFA seminar talk about the hyperplane problem. We add logarithmical estimates concerning the hyperplane conjecture for proportional subspaces and quotients of Banach spaces"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/9312208","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"}