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Given a convex body\n K<\/jats:italic>\n in\n \n \n $$\\mathbb {R}^d$$<\/jats:tex-math>\n \n \n \n R<\/mml:mi>\n <\/mml:mrow>\n d<\/mml:mi>\n <\/mml:msup>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n for fixed\n d<\/jats:italic>\n , the objective is to minimize the number of facets of an approximating polytope for a given Hausdorff error\n \n \n $$\\varepsilon $$<\/jats:tex-math>\n \n \u03b5<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n . It is known that\n \n \n $$O(({{\\,\\textrm{diam}\\,}}(K)\/\\varepsilon )^{(d-1)\/2})$$<\/jats:tex-math>\n \n \n O<\/mml:mi>\n (<\/mml:mo>\n \n \n (<\/mml:mo>\n \n \n diam<\/mml:mtext>\n \n <\/mml:mrow>\n \n (<\/mml:mo>\n K<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \/<\/mml:mo>\n \u03b5<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \n (<\/mml:mo>\n d<\/mml:mi>\n -<\/mml:mo>\n 1<\/mml:mn>\n )<\/mml:mo>\n \/<\/mml:mo>\n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n facets suffice and are necessary for many instances, such as the Euclidean ball. However, this bound is far from optimal for \u201cskinny\u201d convex bodies. A natural way to characterize the skinniness of a convex object is in terms of its relationship to the Euclidean ball. Given a convex body\n K<\/jats:italic>\n , its\n volume diameter<\/jats:italic>\n \n \n $$\\Delta _d(K)$$<\/jats:tex-math>\n \n \n \n \u0394<\/mml:mi>\n d<\/mml:mi>\n <\/mml:msub>\n \n (<\/mml:mo>\n K<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n is defined to be the diameter of a Euclidean ball of the same volume as\n K<\/jats:italic>\n . The\n surface diameter<\/jats:italic>\n \n \n $$\\Delta _{d-1}(K)$$<\/jats:tex-math>\n \n \n \n \u0394<\/mml:mi>\n \n d<\/mml:mi>\n -<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msub>\n \n (<\/mml:mo>\n K<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n is defined analogously for surface area. It follows from generalizations of the isoperimetric inequality that\n \n \n $${{\\,\\textrm{diam}\\,}}(K) \\ge \\Delta _{d-1}(K) \\ge \\Delta _d(K)$$<\/jats:tex-math>\n \n \n \n \n diam<\/mml:mtext>\n \n <\/mml:mrow>\n \n (<\/mml:mo>\n K<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \u2265<\/mml:mo>\n \n \u0394<\/mml:mi>\n \n d<\/mml:mi>\n -<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msub>\n \n (<\/mml:mo>\n K<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \u2265<\/mml:mo>\n \n \u0394<\/mml:mi>\n d<\/mml:mi>\n <\/mml:msub>\n \n (<\/mml:mo>\n K<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n . Arya, da Fonseca, and Mount proved that the diameter-based bound could be made sensitive to the surface diameter, improving the above bound to\n \n \n $$O((\\Delta _{d-1}(K)\/\\varepsilon )^{(d-1)\/2})$$<\/jats:tex-math>\n \n \n O<\/mml:mi>\n (<\/mml:mo>\n \n \n (<\/mml:mo>\n \n \u0394<\/mml:mi>\n \n d<\/mml:mi>\n -<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msub>\n \n (<\/mml:mo>\n K<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \/<\/mml:mo>\n \u03b5<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \n (<\/mml:mo>\n d<\/mml:mi>\n -<\/mml:mo>\n 1<\/mml:mn>\n )<\/mml:mo>\n \/<\/mml:mo>\n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n . In this paper, we strengthen this by proving the existence of an approximation with\n \n \n $$O((\\Delta _d(K)\/\\varepsilon )^{(d-1)\/2})$$<\/jats:tex-math>\n \n \n O<\/mml:mi>\n (<\/mml:mo>\n \n \n (<\/mml:mo>\n \n \u0394<\/mml:mi>\n d<\/mml:mi>\n <\/mml:msub>\n \n (<\/mml:mo>\n K<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \/<\/mml:mo>\n \u03b5<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \n (<\/mml:mo>\n d<\/mml:mi>\n -<\/mml:mo>\n 1<\/mml:mn>\n )<\/mml:mo>\n \/<\/mml:mo>\n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n facets. As a function of volume alone, this bound is tight up to constant factors. Our improvements arise from a combination of new ideas. We exploit known properties of the original body and its polar dual. In order to obtain a volume-sensitive bound, we explore the problem of computing a low-complexity polytope that is sandwiched between two given convex bodies. We show that this problem can be reduced to a covering problem involving a natural intermediate body based on the harmonic mean. Our proof relies on a geometric analysis of a relative notion of fatness involving these bodies.\n <\/jats:p>","DOI":"10.1007\/s00454-025-00780-z","type":"journal-article","created":{"date-parts":[[2025,9,20]],"date-time":"2025-09-20T18:04:21Z","timestamp":1758391461000},"page":"839-871","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Optimal Volume-Sensitive Bounds for Polytope Approximation"],"prefix":"10.1007","volume":"74","author":[{"given":"Sunil","family":"Arya","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3290-8932","authenticated-orcid":false,"given":"David M.","family":"Mount","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,9,20]]},"reference":[{"key":"780_CR1","doi-asserted-by":"publisher","unstructured":"Abdelkader, A., Mount, D.\u00a0M.: Economical Delone sets for approximating convex bodies. 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