测地空间中集合的相互最近点和相互最远点
Mutually nearest and mutually farthest points of sets in geodesic spaces
摘要：
设$A$和$X$是测地度量空间$(E,d)$的非空有界闭子集.最小化（分别。最大化）问题表示为$\min(A,X)$（分别$\max(A,X)$）在于找到$(a_0,x_0) \in A \times X$，使得$d(a_0,x_0) = \inf\{d(a,x) : a \in A, x \in X\}$（分别$d(a_0,x_0) = \sup\{d(a,x) : a \in A, x \in X\}$）。我们研究了这些问题在不同的测地空间中的适定性。设$P_{b,cl,cv}(E)$是$E$的所有非空、有界、闭、凸子集所构成的空间，并赋予Pompeiu-Hausdorff距离.证明了在具有凸度量、曲率有界和测地扩张性质的空间中，$P_{b,cl,cv}(E)$中$\max(A,X)$适定性的集合族是$P_{b,cl,cv}(E)$中的稠密$G_δ$ -集.对于$\min(A,X)$，我们给出了类似的结果，而不需要测地线扩张性质。此外，我们分析了一个集合或两个集合都是紧的情况，并证明了CAT $(0)$空间的一些特殊结果。我们还证明了一个变种的滴定理在测地空间与凸度量，并应用它获得凸函数的优化结果。
英文摘要：
Let $A$ and $X$ be nonempty, bounded and closed subsets of a geodesic metric space $(E,d)$. The minimization (resp. maximization) problem denoted by $\min(A,X)$ (resp. $\max(A,X)$) consists in finding $(a_0,x_0) \in A \times X$ such that $d(a_0,x_0) = \inf\{d(a,x) : a \in A, x \in X\}$ (resp. $d(a_0,x_0) = \sup\{d(a,x) : a \in A, x \in X\}$). We study the well-posedness of these problems in different geodesic spaces considering the set $A$ fixed. Let $P_{b,cl,cv}(E)$ be the space of all nonempty, bounded, closed and convex subsets of $E$ endowed with the Pompeiu-Hausdorff distance. We show that in a space with a convex metric, curvature bounded below and the geodesic extension property, the family of sets in $P_{b,cl,cv}(E)$ for which $\max(A,X)$ is well-posed is a dense $G_δ$-set in $P_{b,cl,cv}(E)$. We give a similar result for $\min(A,X)$ without needing the geodesic extension property. Besides, we analyze the situations when one set or both sets are compact and prove some results specific to CAT$(0)$ spaces. We also prove a variant of the Drop theorem in geodesic spaces with a convex metric and apply it to obtain an optimization result for convex functions.
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