Improved Krasnosel'skij theorems for the dimension of the kernel of a starshaped set
type: Item
instance of: scholarly article
title: Improved Krasnosel'skij theorems for the dimension of the kernel of a starshaped set
zbMATH Open document ID: 0611.52006
DOI: 10.1007/BF01224554
author: Marilyn Breen
published in: Journal of Geometry
publication date: 1986-01-01T00:00:00Z
review text: All sets used are compact subsets of Euclidean space \(R^ d\). x is visible from y via B iff the segment [x,y] lies in B. x is clearly visible from y via B iff there is some neighborhood N of x such that y sees via B each point of \(N\cap B\). S is locally starshaped iff for each point s in S there is some neighborhood N of s which is starshaped, i.e. there is some point p in N such that p sees via N each point of N. The set of all such points is called the (convex) kernel of N. S is starshaped iff S itself can be taken as the neighborhood N at all points s of S. Point s in S is called a point of local non-convexity (lnc point) of S if S fails to be locally convex at s, i.e. for all neighborhoods N of s, \(N\cap S\) is non-convex. For each k and d, \(0\leq k\leq d\), define \(f(d,k)= d+1\) if \(k=0\) and \(f(d,k)=\max \{d+1,2d-2k+2\}\) if \(1\leq k\leq d\). The author proves: I) If S is compact, connected, locally starshaped and non-convex, then for a k with \(0\leq k\leq d\), \(\dim \ker nel\quad S\geq k\) iff every f(d,k) lnc points of S are clearly visible from a common k- dimensional subset of S. II) If \(S\neq \emptyset\) is compact, then for a k with \(0\leq k\leq d\), \(\dim \ker nel\quad S\geq k\) iff every f(d,k) boundary points of S are clearly visible from a common k-dimensional subset of S. In both cases, the number f(d,k) is best possible for every d and k.
Mathematics Subject Classification ID: 52A30
Mathematics Subject Classification ID: 52A35
zbMATH DE Number: 3988126
zbMATH Keywords: dimension of the kernel of a starshaped set
zbMATH Keywords: Krasnosel'skij theorem
MaRDI profile type: Publication
cites work: Clear Visibility and the Dimension of Kernels of Starshaped Sets
cites work: Increasing unions of starlike sets
cites work: Points of local nonconvexity, clear visibility, and starshaped sets in \(R^ n\)
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description: scientific article; zbMATH DE number 3988126