庞贝问题

摘要：

令$f \in L_{loc}^1 (\R^n)\cap \mathcal{S}'$，其中$\mathcal{S}'$是Schwartz分布类，和$$\int_{σ(D)} f(x) dx = 0 \quad \forall σ\in G, \qquad (*)$$，其中$D\subset \R^n$是有界区域，其闭包$\bar{D}$同构于闭球。然后连接$\bar{D}$的补码并连接路径。这里$G$表示$\R^n$中所有刚体运动的组。该组由所有平移和旋转组成。假设$f\neq 0$和（*）成立，则$D$是球。其他的结构，相当于上述之一，制定和讨论。对已有的结果给出了几个新的简短证明。

英文摘要：

Let $f \in L_{loc}^1 (\R^n)\cap \mathcal{S}'$, where $\mathcal{S}'$ is the Schwartz class of distributions, and $$\int_{σ(D)} f(x) dx = 0 \quad \forall σ\in G, \qquad (*)$$ where $D\subset \R^n$ is a bounded domain, the closure $\bar{D}$ of which is diffeomorphic to a closed ball. Then the complement of $\bar{D}$ is connected and path connected. Here $G$ denotes the group of all rigid motions in $\R^n$. This group consists of all translations and rotations. It is conjectured that if $f\neq 0$ and (*) holds, then $D$ is a ball. Other conjectures, equivalent to the above one, are formulated and discussed. Several new short proofs are given for the earlier proved results.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。