{"paper":{"title":"Extremal Relations Between Shannon Entropy and $\\ell_{\\alpha}$-Norm","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.IT"],"primary_cat":"cs.IT","authors_text":"Ken-ichi Iwata, Yuta Sakai","submitted_at":"2016-01-28T07:56:38Z","abstract_excerpt":"The paper examines relationships between the Shannon entropy and the $\\ell_{\\alpha}$-norm for $n$-ary probability vectors, $n \\ge 2$. More precisely, we investigate the tight bounds of the $\\ell_{\\alpha}$-norm with a fixed Shannon entropy, and vice versa. As applications of the results, we derive the tight bounds between the Shannon entropy and several information measures which are determined by the $\\ell_{\\alpha}$-norm, e.g., R\\'{e}nyi entropy, Tsallis entropy, the $R$-norm information, and some diversity indices. Moreover, we apply these results to uniformly focusing channels. Then, we show"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.07678","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"}