{"paper":{"title":"Entropy of convex functions on $R^d$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["stat.TH"],"primary_cat":"math.ST","authors_text":"Fuchang Gao, Jon A. Wellner","submitted_at":"2015-02-05T22:46:08Z","abstract_excerpt":"Let $\\Omega$ be a bounded closed convex set in ${\\mathbb R}^d$ with non-empty interior, and let ${\\cal C}_r(\\Omega)$ be the class of convex functions on $\\Omega$ with $L^r$-norm bounded by $1$. We obtain sharp estimates of the $\\epsilon$-entropy of ${\\cal C}_r(\\Omega)$ under $L^p(\\Omega)$ metrics, $1\\le p<r\\le \\infty$. In particular, the results imply that the universal lower bound $\\epsilon^{-d/2}$ is also an upper bound for all $d$-polytopes, and the universal upper bound of $\\epsilon^{-\\frac{(d-1)}{2}\\cdot \\frac{pr}{r-p}}$ for $p>\\frac{dr}{d+(d-1)r}$ is attained by the closed unit ball. Whi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1502.01752","kind":"arxiv","version":3},"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"}