Improved Krasnosel'skij theorems for the dimension of the kernel of a starshaped set
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.
dimension of the kernel of a starshaped set
scientific article; zbMATH DE number 3988126