由水平集包围的区域的体积的可微性
Differentiability of the volume of a region enclosed by level sets
摘要：
函数f在n维空间上的水平包含一个区域。两个这样的水平之间的区域的体积取决于这两个水平。固定其中之一，体积成为剩余水平的函数。我们证明，如果函数f是光滑的，体积函数也是光滑的，对于f的正则值。对于f的临界值，体积函数仅是可微的。这项研究的最初动机来自放射治疗，其中这种体积函数用于优化过程。因此，它们的可微性变得很重要。
英文摘要：
The level of a function f on an n-dimensional space encloses a region. The volume of a region between two such levels depends on both levels. Fixing one of them the volume becomes a function of the remaining level. We show that if the function f is smooth, the volume function is again smooth for regular values of f. For critical values of f the volume function is only finitely differentiable. The initial motivation for this study comes from Radiotherapy, where such volume functions are used in an optimization process. Thus their differentiability properties become important.
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