Carnot群中的等径不等式
Isodiametric inequality in Carnot groups
摘要：
欧几里得空间中经典的等直径不等式说，球在具有给定直径的所有集合中使体积最大化。本文考虑卡诺群的情形。我们证明，任何卡诺集团配备了哈尔措施，可以找到一个齐次的距离，这未能举行。我们还考虑卡诺-Caratheodory距离，并证明这也失败了这些距离，只要有长度最小化曲线，停止在有限时间内最小化。接下来，我们研究了一些连接与比较Hausdorff和球形Hausdorff措施，rectifiability和广义1/2-Besicovitch猜想，特别是在某些情况下，这个猜想失败。
英文摘要：
The classical isodiametric inequality in the Euclidean space says that balls maximize the volume among all sets with a given diameter. We consider in this paper the case of Carnot groups. We prove that for any Carnot group equipped with a Haar measure one can find a homogeneous distance for which this fails to hold. We also consider Carnot-Caratheodory distances and prove that this also fails for these distances as soon as there are length minimizing curves that stop to be minimizing in finite time. Next we study some connections with the comparison between Hausdorff and spherical Hausdorff measures, rectifiability and the generalized 1/2-Besicovitch conjecture giving in particular some cases where this conjecture fails.
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