关于凸扩张的连续性和正则性
On the continuity and regularity of convex extensions
摘要：
研究了从紧集$C$到其凸包$K$的函数凸扩张的连续性和正则性。证明了如果$C$包含$K$的相对边界，且$f$是$C$上的连续凸函数，则$f$利用标准凸屋顶构造扩展为$K$上的连续凸函数.事实上，$f$从任意集合扩张到凸包上的连续凸函数的充要条件是$f$扩张到凸包的相对边界上的连续凸函数.我们给出的例子表明，结果中的假设是必要的。特别地，如果$C$不包含$K$的整个相对边界，则可能不存在$f$的任何连续凸扩展。最后，当$K$和$f$的边界为$C^1$时，给出了凸屋顶构造为所有$K$上的$C^1$的充要条件.我们还讨论了凸屋顶结构在量子计算中的应用。
英文摘要：
We study continuity and regularity of convex extensions of functions from a compact set $C$ to its convex hull $K$. We show that if $C$ contains the relative boundary of $K$, and $f$ is a continuous convex function on $C$, then $f$ extends to a continuous convex function on $K$ using the standard convex roof construction. In fact, a necessary and sufficient condition for $f$ to extend from any set to a continuous convex function on the convex hull is that $f$ extends to a continuous convex function on the relative boundary of the convex hull. We give examples showing that the hypotheses in the results are necessary. In particular, if $C$ does not contain the entire relative boundary of $K$, then there may not exist any continuous convex extension of $f$. Finally, when the boundary of $K$ and $f$ are $C^1$ we give a necessary and sufficient condition for the convex roof construction to be $C^1$ on all of $K$. We also discuss an application of the convex roof construction in quantum computation.
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