{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T03:46:36Z","timestamp":1740109596299,"version":"3.37.3"},"reference-count":27,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2023,4,19]],"date-time":"2023-04-19T00:00:00Z","timestamp":1681862400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2023,4,19]],"date-time":"2023-04-19T00:00:00Z","timestamp":1681862400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Discrete Comput Geom"],"published-print":{"date-parts":[[2023,7]]},"abstract":"Abstract<\/jats:title>Let K<\/jats:italic> be a centrally symmetric spherical and simplicial polytope, whose vertices form a $$(4n)^{-1}$$<\/jats:tex-math>\n \n \n (<\/mml:mo>\n 4<\/mml:mn>\n n<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \n -<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-net in the unit sphere in\u00a0$${\\mathbb R}^n$$<\/jats:tex-math>\n \n \n R<\/mml:mi>\n <\/mml:mrow>\n n<\/mml:mi>\n <\/mml:msup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. We prove a uniform lower bound on the norms of all hyperplane projections $$P:X\\rightarrow X$$<\/jats:tex-math>\n \n P<\/mml:mi>\n :<\/mml:mo>\n X<\/mml:mi>\n \u2192<\/mml:mo>\n X<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, where X<\/jats:italic> is the n<\/jats:italic>-dimensional normed space with the unit ball\u00a0K<\/jats:italic>. The estimate is given in terms of the determinant function of vertices and faces of\u00a0K<\/jats:italic>. In particular, if $$N\\ge n^{4n}$$<\/jats:tex-math>\n \n N<\/mml:mi>\n \u2265<\/mml:mo>\n \n n<\/mml:mi>\n \n 4<\/mml:mn>\n n<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and $$K={{\\,\\textrm{conv}\\,}}{\\{\\pm x_1,\\pm x_2,\\dots ,\\pm x_N\\}}$$<\/jats:tex-math>\n \n K<\/mml:mi>\n =<\/mml:mo>\n \n \n conv<\/mml:mtext>\n \n <\/mml:mrow>\n \n {<\/mml:mo>\n \u00b1<\/mml:mo>\n \n x<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n ,<\/mml:mo>\n \u00b1<\/mml:mo>\n \n x<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msub>\n ,<\/mml:mo>\n \u22ef<\/mml:mo>\n ,<\/mml:mo>\n \u00b1<\/mml:mo>\n \n x<\/mml:mi>\n N<\/mml:mi>\n <\/mml:msub>\n }<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, where $$x_1,x_2,\\dots ,x_N$$<\/jats:tex-math>\n \n \n x<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n ,<\/mml:mo>\n \n x<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msub>\n ,<\/mml:mo>\n \u22ef<\/mml:mo>\n ,<\/mml:mo>\n \n x<\/mml:mi>\n N<\/mml:mi>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> are independent random points distributed uniformly in the unit sphere, then every hyperplane projection $$P:X \\rightarrow X$$<\/jats:tex-math>\n \n P<\/mml:mi>\n :<\/mml:mo>\n X<\/mml:mi>\n \u2192<\/mml:mo>\n X<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> satisfies an inequality $$\\Vert P\\Vert _X\\ge 1+c_nN^{-(2n^2+4n+6)}$$<\/jats:tex-math>\n \n \n \n \u2016<\/mml:mo>\n P<\/mml:mi>\n \u2016<\/mml:mo>\n <\/mml:mrow>\n X<\/mml:mi>\n <\/mml:msub>\n \u2265<\/mml:mo>\n 1<\/mml:mn>\n +<\/mml:mo>\n \n c<\/mml:mi>\n n<\/mml:mi>\n <\/mml:msub>\n \n N<\/mml:mi>\n \n -<\/mml:mo>\n (<\/mml:mo>\n 2<\/mml:mn>\n \n n<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n +<\/mml:mo>\n 4<\/mml:mn>\n n<\/mml:mi>\n +<\/mml:mo>\n 6<\/mml:mn>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> (for some explicit constant\u00a0$$c_n$$<\/jats:tex-math>\n \n c<\/mml:mi>\n n<\/mml:mi>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>), with the probability at least $$1-3\/N$$<\/jats:tex-math>\n \n 1<\/mml:mn>\n -<\/mml:mo>\n 3<\/mml:mn>\n \/<\/mml:mo>\n N<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>.<\/jats:p>","DOI":"10.1007\/s00454-023-00506-z","type":"journal-article","created":{"date-parts":[[2023,4,19]],"date-time":"2023-04-19T14:03:27Z","timestamp":1681913007000},"page":"279-296","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A Uniform Lower Bound on the Norms of Hyperplane Projections of Spherical Polytopes"],"prefix":"10.1007","volume":"70","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1571-5549","authenticated-orcid":false,"given":"Tomasz","family":"Kobos","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2023,4,19]]},"reference":[{"key":"506_CR1","doi-asserted-by":"publisher","first-page":"53","DOI":"10.1007\/BF01766140","volume":"152","author":"M Baronti","year":"1988","unstructured":"Baronti, M., Papini, P.L.: Norm-one projections onto subspaces of $$l_p$$. 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