关于WDC集的结构
On the structure of WDC sets
摘要：
${\mathbb R}^d$中的WDC集最近被定义为DC函数（凸函数的差）在弱正则值处的子级集。他们形成了一个自然的和实质性的推广集的积极达到，仍然承认的定义曲率措施。利用凸函数的奇异性结果，得到了WDC集边界上的正则性结果。特别地，紧WDC集的边界可以被许多DC曲面覆盖。更一般地，我们证明了任何紧WDC集$M$的拓扑维数$k\leq d$可以分解成两个集的并集，其中一个是$k$维DC流形开放的$M$，另一个可以覆盖的DC曲面的维数$k-1$。我们还刻画了局部WDC集之间的封闭Lipschitz域和低维Lipschitz流形。最后，我们找到了平面上局部WDC集的一个完整刻画。
英文摘要：
WDC sets in ${\mathbb R}^d$ were recently defined as sublevel sets of DC functions (differences of convex functions) at weakly regular values. They form a natural and substantial generalization of sets with positive reach and still admit the definition of curvature measures. Using results on singularities of convex functions, we obtain regularity results on the boundaries of WDC sets. In particular, the boundary of a compact WDC set can be covered by finitely many DC surfaces. More generally, we prove that any compact WDC set $M$ of topological dimension $k\leq d$ can be decomposed into the union of two sets, one of them being a $k$-dimensional DC manifold open in $M$, and the other can be covered by finitely many DC surfaces of dimension $k-1$. We also characterize locally WDC sets among closed Lipschitz domains and among lower-dimensional Lipschitz manifolds. Finally, we find a full characterization of locally WDC sets in the plane.
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