On the Christoffel-Minkowski problem of Firey’s p-sum
On the Christoffel-Minkowski problem of Firey’s p-sum
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DOI：
10.1007/s00526-003-0250-9
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发表时间：
2004-10
影响因子：
2.1
通讯作者：
Changqing Hu;Xinan Ma;C. Shen
中科院分区：
文献类型：
--
作者：
Changqing Hu;Xinan Ma;C. Shen
经典的Brunn-Minkowski凸体理论是从几个基本概念发展而来的：支撑函数、Minkowski组合和混合体。作为混合体的特例，Quermass积分是凸体的重要几何量，而表面积度量是Quermasals的局部形式。Christoffel-Minkowski问题涉及具有指定表面积度量的凸体的存在性，具体内容请参考[17，3，8，18]。1962年，Firey[5]将Minkowski组合从p=1推广到p-≥1。后来，Lutwak[13，14]证明了对每个p>；1，Firey的p-和也导致了Brunn-Minkowski理论。这个理论已经发现了许多几何应用，例如，参见[16]及其参考文献。文献[13]还表明，经典的表面积度量可以推广到p-和的情形。因此，对于每个p>；1，考虑经典的Christoffel-Minkowski问题的推广是很自然的。文献[13，15，7，4]中已经讨论了推广的Minkowski问题。在本文中，我们研究剩余的情形，它可以称为p-和的Christoffel-Minkowski问题。首先，我们介绍了一些记号和相关结果。
The classical Brunn-Minkowski theory for convex bodies was developed from a few basic concepts: support functions, Minkowski combinations, and mixed volumes. As a special case of mixed volumes, the Quermassintegrals are important geometrical quantities of a convex body, and surface area measures are local versions of Quermassintegrals. The Christoffel-Minkowski problem concerns with the existence of convex bodies with prescribed surface area measure, for details please refer to [17, 3, 8, 18].In 1962, Firey [5] generalized the Minkowski combination to p-sums from p= 1 to p≥ 1. Later, Lutwak [13, 14] showed that Firey’s p-sum also leads to a Brunn-Minkowski theory for each p> 1. This theory has found many geometry applications, see for example,[16] and its references. It was also shown in [13] that the classical surface area measures could be extended to the p-sum case. So it is natural to consider a generalization of the classical Christoffel-Minkowski problem for each p> 1. The generalized Minkowski problem has been treated in [13, 15, 7, 4]. In this paper we study the remaining case, which may be called the Christoffel-Minkowski problem of p-sum. First we introduce some notations and relevant results.