论文标题
表征凸形域的不平等
An inequality characterizing convex domains
论文作者
论文摘要
光滑凸域的属性$ω\ subset \ mathbb {r}^n $是，如果边界上有两个点$ x，y \ in \ partialω$彼此接近，则它们的普通向量$ n（x），n（y），n（y）$大致相同的方向，而这个方向几乎是$ x-y $ y $ y $ y $ y $ x $ x $ x $ x $ x和$ x $ x $ x $ x $ x $ x $ x $ x和x $ x $ x $ x $ x $ x和$ x $ x和$ x $ x $ x $ x $ x和x $ x.我们证明存在常数$ C_N> 0 $，因此，如果$ω\ subset \ mathbb {r}^n $是一个有界域，则具有$ c^1- $ boundard $ \ partialω$，则$ c^1- $ \ int \ int \ int _ {\ int _ {\ partialω\ partialω\ times \ partialω}} \ frac \ eft y \左\ right \ rangle \ left \ langle y -x，n（y）\ right \ rangle \ right | } {\ | x -y \ |^{n+1}} 〜dσ（x）dσ（y）\ geq c_n | \ partialω| $ $ $ $ and equality and equality and equaly在且仅当域$ω$ is convex时发生。
A property of smooth convex domains $Ω\subset \mathbb{R}^n$ is that if two points on the boundary $x, y \in \partial Ω$ are close to each other, then their normal vectors $n(x), n(y)$ point roughly in the same direction and this direction is almost orthogonal to $x-y$ (for `nearby' $x$ and $y$). We prove there exists a constant $c_n > 0$ such that if $Ω\subset \mathbb{R}^n$ is a bounded domain with $C^1-$boundary $\partial Ω$, then $$ \int_{\partial Ω\times \partial Ω} \frac{\left|\left\langle n(x), y - x \right\rangle \left\langle y - x, n(y) \right\rangle \right| }{\|x - y\|^{n+1}}~d σ(x) dσ(y) \geq c_n |\partial Ω|$$ and equality occurs if and only if the domain $Ω$ is convex.
