Radial continuous rotation invariant valuations on star bodies

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• 作者：Ignacio Villanueva ^{ignaciov@mat.ucm.es" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Convex geometry ; Valuations ; Star bodies
• 刊名：Advances in Mathematics
• 出版年：2016
• 出版时间：19 March 2016
• 年：2016
• 卷：291
• 期：Complete
• 页码：961-981
• 全文大小：412 K

文摘

We characterize the positive radial continuous and rotation invariant valuations V   defined on the star bodies of formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816000335&_mathId=si1.gif&_user=111111111&_pii=S0001870816000335&_rdoc=1&_issn=00018708&md5=1009fc3815b7f749e241776c71a1bf64" title="Click to view the MathML source">Rⁿflow="scroll">Rn as the applications on star bodies which admit an integral representation with respect to the Lebesgue measure. That is, where θ   is a positive continuous function, formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816000335&_mathId=si17.gif&_user=111111111&_pii=S0001870816000335&_rdoc=1&_issn=00018708&md5=11ee68445d20101514f6ad61db986c10" title="Click to view the MathML source">ρ_Kflow="scroll">ρK is the radial function associated to K and m   is the Lebesgue measure on formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816000335&_mathId=si16.gif&_user=111111111&_pii=S0001870816000335&_rdoc=1&_issn=00018708&md5=bef5f687a25c54fd3cbfb02302288206" title="Click to view the MathML source">Sⁿ⁻¹flow="scroll">Sn−1. As a corollary, we obtain that every such valuation can be uniformly approximated on bounded sets by a linear combination of dual quermassintegrals.