论文标题
弯曲度量和肥皂气泡超出了凸度
Curvature measures and soap bubbles beyond convexity
论文作者
论文摘要
Extending the celebrated results of Alexandrov (1958) and Korevaar-Ros (1988) for smooth sets, as well as the results of Schneider (1979) and the first author (1999) for arbitrary convex bodies, we obtain for the first time the characterization of the isoperimetric sets of a uniformly convex smooth finite-dimensional normed space (i.e. Wulff shapes) in the non-smooth基于涉及曲率度量的自然几何条件和非凸面设置。更具体地说，我们在自然的平均值假设下表明，有限的不相交的沃尔夫形状是唯一的积极到达$ a \ subseteq \ subseteq \ subseteq \ mathbf {r}^{n+1} $具有有限和正数的$ n $ k \ in \ in \ k \ k $ ndots， θ^ϕ_ {k}（a，\ cdot）$，在$ a $ a $的单位捆绑上定义相对于由$ ϕ $引起的相对几何形状，与$θ^ϕ_ {n}（a，\ cdot）$成正比。如果$ k = n-1 $，结论对于所有正面覆盖率的结论都具有有限和正数。我们还证明了有关奇异性的可移动性的相关结果。该结果基于Busemann and Feller（1936）用于任意凸体的正常边界点的概念的扩展，以延伸到正覆盖范围。 即使在欧几里得空间中，这些结果也是新的。 还证明了几个辅助和相关的结果，它们具有独立的兴趣。其中包括在有限的维度均匀凸的范围矢量空间和局部体积函数导数的一般公式中的任意封闭套件的延伸到任意封闭的集合，从而扩展并补充了Chambolle-lussardi-lussardi-villa（2021）的最新结果。
Extending the celebrated results of Alexandrov (1958) and Korevaar-Ros (1988) for smooth sets, as well as the results of Schneider (1979) and the first author (1999) for arbitrary convex bodies, we obtain for the first time the characterization of the isoperimetric sets of a uniformly convex smooth finite-dimensional normed space (i.e. Wulff shapes) in the non-smooth and non-convex setting, based on the natural geometric condition involving the curvature measures. More specifically we show, under a natural mean-convexity assumption, that finite unions of disjoint Wulff shapes are the only sets of positive reach $ A \subseteq \mathbf{R}^{n+1} $ with finite and positive volume such that, for some $ k \in \{0, \ldots , n-1\}$, the $ k $-th generalized curvature measure $ Θ^ϕ_{k}(A, \cdot) $, which is defined on the unit normal bundle of $ A $ with respect to the relative geometry induced by $ ϕ$, is proportional to $ Θ^ϕ_{n}(A, \cdot)$. If $ k = n-1 $ the conclusion holds for all sets of positive reach with finite and positive volume. We also prove a related sharp result about the removability of the singularities. This result is based on the extension of the notion of a normal boundary point, originally introduced by Busemann and Feller (1936) for arbitrary convex bodies, to sets of positive reach. These results are new even in the Euclidean space. Several auxiliary and related results are also proved, which are of independent interest. They include the extension of the classical Steiner--Weyl tube formula to arbitrary closed sets in a finite dimensional uniformly convex normed vector space and a general formula for the derivative of the localized volume function, which extends and complements recent results of Chambolle--Lussardi--Villa (2021).
