论文标题
内部和外部平滑近似的凸出曲面。什么时候可能？
Inner and outer smooth approximation of convex hypersurfaces. When is it possible?
论文作者
论文摘要
令$ s $为$ \ mathbb {r}^n $中的凸出hypersurface（封闭凸面$ v $的边界）。我们证明$ s $不包含任何行，并且仅当每个开放设置$ u \ u \ supset s $中都存在一个实用分析的凸hypersurface $ s_ {u} \ subset u \ cap \ cap \ cap \ textrm {int}（v）$。我们还表明，$ s $在每个开放式$ u \ supset s $中都没有射线，存在一个真实的分析凸出hypersurface $ s_ {u} \ subset u \ setMinus v $。此外，在这两种情况下，$ S_U $都可以强烈凸出。我们还为在$ \ mathbb {r}^n $的开放凸子集上定义的凸函数建立了相似的结果，完全表征了凸函数类别可以通过$ c^0 $ fine拓扑在上面或下方的平滑凸函数中近似的凸函数。我们还为$ c^1 $ fine近似值提供了类似的结果
Let $S$ be a convex hypersurface (the boundary of a closed convex set $V$ with nonempty interior) in $\mathbb{R}^n$. We prove that $S$ contains no lines if and only if for every open set $U\supset S$ there exists a real-analytic convex hypersurface $S_{U} \subset U\cap \textrm{int}(V) $. We also show that $S$ contains no rays if and only if for every open set $U\supset S$ there exists a real-analytic convex hypersurface $S_{U}\subset U\setminus V$. Moreover, in both cases, $S_U$ can be taken strongly convex. We also establish similar results for convex functions defined on open convex subsets of $\mathbb{R}^n$, completely characterizing the class of convex functions that can be approximated in the $C^0$-fine topology by smooth convex functions from above or from below. We also provide similar results for $C^1$-fine approximations
