论文标题
在有限维规范的空间超过超级空间中的Fermat-稳定性问题中的边界稳定性
Boundary stability in the Fermat--Steiner problem in hyperspaces over finite-dimensional normed spaces
论文作者
论文摘要
FERMAT - steiner问题是找到度量空间的所有点，以便从每个固定有限的子集a = {a_1，...，...，a_n}的距离的总和至最小。当y = h（x）是有限维规范空间x的非空态子集的空间时，考虑了这个问题。在本文中，我们研究了Fermat中的稳定性问题 - Steiner问题，从由有限的紧凑型组组成的边界传递到由凸壳组成的边界。通过稳定性，我们的意思是，距离之和的最小值在传递到边界紧凑型集合的凸壳时不会改变。该论文继续研究了几何物体，即在fermat-steiner问题中出现的钩子集。获得的结果揭示了施纳式紧凑和边界组之间关系的一些几何形状。在其基础上，得出了H（X）中边界不稳定的足够条件。
The Fermat--Steiner problem is to find all points of the metric space Y such that the sum of the distances from each of them to points from some fixed finite subset A = {A_1, ..., A_n} of the space Y is minimal. This problem is considered in the case when Y=H(X) is the space of non-empty compact subsets of a finite-dimensional normed space X endowed with the Hausdorff metric, i.e. H(X) is a hyperspace over X. The set A is called boundary, all A_i are called boundary sets, and the compact sets that realize the minimum of the sum of distances to A_i are called Steiner compacts. In this paper, we study the question of stability in the Fermat--Steiner problem when passing from a boundary consisting of finite compact sets to a boundary consisting of their convex hulls. By stability here we mean that the minimum of the sum of distances does not change when passing to convex hulls of boundary compact sets. The paper continued the study of geometric objects, namely, hook sets that arise in the Fermat--Steiner problem. The results obtained revealed some geometry of the relationship between Steiner compacts and boundary sets. On its basis, a sufficient condition for the instability of the boundary in H(X) was derived.
