论文标题
通过覆盖范围和体积来表征欧几里得空间中的单位球
Characterizing unit spheres in Euclidean spaces via reach and volume
论文作者
论文摘要
令$ m $为$ \ mathbb {r}^n $的平滑，连接，紧凑的submanifold，没有边界和尺寸$ k \ geq 2 $。令$ \ mathbb {s}^k \ subset \ mathbb {r}^{k+1} \ subset \ subset \ mathbb {r}^n $表示$ k $ - dimesnional单位sphere。我们显示$ m $的达到一个等于1，那么它的音量满足$ \ text {vol}（m）\ geq \ text {vol}（\ mathbb {s}^k）$，仅当$ m $与$ \ $ \ mathbb {s}^k $一致时，只有$ m $仅在$ m $的情况下。
Let $M$ be a smooth, connected, compact submanifold of $\mathbb{R}^n$ without boundary and of dimension $k\geq 2$. Let $\mathbb{S}^k \subset \mathbb{R}^{k+1}\subset \mathbb{R}^n$ denote the $k$-dimesnional unit sphere. We show if $M$ has reach equal to one, then its volume satisfies $\text{vol}(M)\geq \text{vol}(\mathbb{S}^k)$ with equality holding only if $M$ is congruent to $\mathbb{S}^k$.
