论文标题
凸函数和凸体的Lusin型特性
Lusin-type properties of convex functions and convex bodies
论文作者
论文摘要
我们证明，如果$ f：\ mathbb {r}^n \ to \ mathbb {r} $为凸，$ a \ subset \ subset \ mathbb {r}^n $具有有限的度量，那么对于任何$ \ \ varepsilon> 0 $，有一个convex函数$ g：\ mathbb {rmathbb {r} $ {r} $ {r} $ c^{1,1} $使得$ \ Mathcal {l}^n（\ {x \ in A：\，f（x）\ neq g（x）\}）<\ varepsilon $。作为一个应用程序，我们推断出，如果$ w \ subset \ mathbb {r}^n $是一个紧凑的凸构体，那么，对于每一个$ \ varepsilon> 0 $，都存在一个convex $ w _ {\ varepsilon} class $ c^{1,1} $ c^{1,1} $ \ \ m nath的class $ c^{ w \ setMinus \ partial w _ {\ varepsilon} \ right）<\ varepsilon $。我们还表明，如果$ f：\ mathbb {r}^n \ to \ mathbb {r} $是convex函数，而$ f $不是类$ c^{1,1} _ {\ rm loc} $，那么对于任何$ \ varepsilon> 0 $ is convex函数$ g：\ mathbb {r}^n \ to \ mathbb {r} $ of类$ c^{1,1} _ {\ rm loc} $，以至于$ \ mathcal {l}^n（\ r} g（x）\}）<\ varepsilon $，仅当$ f $本质上是强制性的，这意味着$ \ lim_ {| x | \ to \ infty} f（x） - \ ell（x） - \ ell（x）= \ infty $用于某些线性功能$ \ ell $。结果的结果是，如果$ s $是$ \ mathbb {r}^n $和$ s $的某些凸面的边界（不必$ c^{1,1} _ {\ textrm {loc}} $，以便$ \ mathcal {h}^{n-1}（s \ setminus s _ {\ varepsilon}）<\ varepsilon $。
We prove that if $f:\mathbb{R}^n\to\mathbb{R}$ is convex and $A\subset\mathbb{R}^n$ has finite measure, then for any $\varepsilon>0$ there is a convex function $g:\mathbb{R}^n\to\mathbb{R}$ of class $C^{1,1}$ such that $\mathcal{L}^n(\{x\in A:\, f(x)\neq g(x)\})<\varepsilon$. As an application we deduce that if $W\subset\mathbb{R}^n$ is a compact convex body then, for every $\varepsilon>0$, there exists a convex body $W_{\varepsilon}$ of class $C^{1,1}$ such that $\mathcal{H}^{n-1}\left(\partial W\setminus \partial W_{\varepsilon}\right)< \varepsilon$. We also show that if $f:\mathbb{R}^n\to\mathbb{R}$ is a convex function and $f$ is not of class $C^{1,1}_{\rm loc}$, then for any $\varepsilon>0$ there is a convex function $g:\mathbb{R}^n\to\mathbb{R}$ of class $C^{1,1}_{\rm loc}$ such that $\mathcal{L}^n(\{x\in \mathbb{R}^n:\, f(x)\neq g(x)\})<\varepsilon$ if and only if $f$ is essentially coercive, meaning that $\lim_{|x|\to\infty}f(x)-\ell(x)=\infty$ for some linear function $\ell$. A consequence of this result is that, if $S$ is the boundary of some convex set with nonempty interior (not necessarily bounded) in $\mathbb{R}^n$ and $S$ does not contain any line, then for every $\varepsilon>0$ there exists a convex hypersurface $S_{\varepsilon}$ of class $C^{1,1}_{\textrm{loc}}$ such that $\mathcal{H}^{n-1}(S\setminus S_{\varepsilon})<\varepsilon$.
