Volume Inequalities for Subspaces of L p
Volume Inequalities for Subspaces of L p
复制标题
DOI：
10.4310/jdg/1102536713
复制
发表时间：
2004-09
影响因子：
2.5
通讯作者：
E. Lutwak;Deane Yang;Gaoyong Zhang
中科院分区：
文献类型：
--
作者：
E. Lutwak;Deane Yang;Gaoyong Zhang
利用直接方法建立了Lp的子空间的单位球的Ball和Barthe逆等周不等式。这种方法的优点是它完全解决了这些不等式的所有开放的唯一性问题。仿射等周不等式通常以椭球为极值。所谓的逆仿射等周不等式通常有单形-或在对称情况下立方体和它们的极-作为它们的极值。世纪前Steiner发展和推广的对称化技术已被用来建立各种强大的仿射等周不等式。相反的不平等将变得更加难以确立。它们似乎需要某种反对称化技术。到1990年，只有一个重要的反向不等式在大于2的维度上成立：Rogers-Shephard差分-体不等式（见例[42]）。不幸的是，罗杰斯和谢泼德所采用的技术不能适用于建立任何其他的逆不等式。一个突破发生在1990年，当基思球连接约翰的定理表征最大的椭球包含在一个凸体（约翰椭球）与Brascamp-Lieb不等式。Brascamp-Lieb不等式是为了解决Young卷积不等式的最佳常数问题而开发的（参见Gardner最近的优秀调查[11]）。鲍尔发现了Brascamp-Lieb不等式的一个华丽的重新表述，似乎是为利用约翰椭球量身定制的。Ball的标准化Brascamp-Lieb不等式对凸几何分析产生了深远的影响（参见，例如，[1，2，3]，[5]，[12]，[13]，[14]，[40]，[41]）。研究部分由NSF Grant DMS-0104363支持。2003年5月12日收到。
A direct approach is used to establish both Ball and Barthe’s reverse isoperimetric inequalities for the unit balls of subspaces of Lp. This approach has the advantage that it completely settles all the open uniqueness questions for these inequalities. Affine isoperimetric inequalities generally have ellipsoids as extremals. The so called reverse affine isoperimetric inequalities usually have simplices – or in the symmetric case cubes and their polars – as their extremals. Symmetrization techniques, developed and promoted by Steiner well over a century ago, have been used to establish a variety of powerful affine isoperimetric inequalities. The reverse inequalities would turn out to be much harder to establish. They appeared to require some sort of antisymmetrization technique. By 1990, only one significant reverse inequality had been established in dimensions greater than two: the Rogers–Shephard difference-body inequality (see e.g. [42]). Unfortunately, the techniques employed by Rogers and Shephard could not be adapted to establish any of the other conjectured reverse inequalities. A breakthrough occurred in 1990, when Keith Ball connected John’s theorem characterizing the largest ellipsoid contained in a convex body (the John ellipsoid) with the Brascamp– Lieb inequality. The Brascamp–Lieb inequality had been developed to solve the best-constant problem for Young’s convolution inequality (see the excellent recent survey of Gardner [11]). Ball discovered a gorgeous reformulation of the Brascamp–Lieb inequality that seemed tailor-made to exploit the John ellipsoid. Ball’s normalized Brascamp– Lieb inequality has had a profound impact on convex geometric analysis (see, e.g., [1, 2, 3], [5], [12], [13], [14], [40], [41]). Research supported, in part, by NSF Grant DMS–0104363. Received 12/05/2003.