The dual Minkowski problem for symmetric convex bodies
The dual Minkowski problem for symmetric convex bodies
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DOI：
10.1016/j.aim.2019.106805
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发表时间：
2017-03
影响因子：
1.7
通讯作者：
K. Boroczky;E. Lutwak;Deane Yang;Gaoyong Zhang;Yiming Zhao
中科院分区：
文献类型：
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作者：
K. Boroczky;E. Lutwak;Deane Yang;Gaoyong Zhang;Yiming Zhao
偶数数据的对偶闵可夫斯基问题询问单位球面上规定的偶数测度作为 R n 中原点对称凸体的第 q 个对偶曲率测度的必要和充分条件。当 1< q< n 时给出该问题的完整解。充分必要条件是显式测度浓度条件。为了获得结果，使用变分方法，其中泛函是对偶质量积分和熵积分的总和。证明需要两个关键的估计。第一个是通过使用球形分区获得的熵积分的估计。第二个是对精心选择的势垒凸体的对偶质量积分的精确估计。
The dual Minkowski problem for even data asks what are the necessary and sufficient conditions on a prescribed even measure on the unit sphere for it to be the q-th dual curvature measure of an origin-symmetric convex body in R n. A full solution to this is given when 1< q< n. The necessary and sufficient conditions turn out to be an explicit measure concentration condition. To obtain the results, a variational approach is used, where the functional is the sum of a dual quermassintegral and an entropy integral. The proof requires two crucial estimates. The first is an estimate of the entropy integral which is obtained by using a spherical partition. The second is a sharp estimate of the dual quermassintegrals for a carefully chosen barrier convex body.