Solution of the problem of combinatorial characterization of the dimension of the kernel of a starshaped set (Q1895159)

In a real linear space, a nonempty subset \(S\) is starshaped if there exists a point \(x\) in \(S\) such that for any point \(y\) in \(S\), the closed line segment connecting \(x\) and \(y\) lies completely in \(S\). The set of all such points \(x\) is the kernel of \(S\), denoted by \(\ker S\). It is proved that \(\ker S = \cap \{\text{conv} A_z : z \in \text{bdry} S\}\) and if \(S\) is closed and connected, then \(\ker S = \cap \{\text{conv} A_z : z \in \text{slnc} S\}\), where \(\text{bdry} S\) and \(\text{slnc} S\) denote the sets of boundary points and strong local nonconvex points of \(S\), respectively, and \(A_z\) is the set of points \(y\) in \(S\) for which each point in some neighborhood of \(z\) is in the kernel. This yields two Krasnosel'skii-type characterizations for the dimension of \(\ker S\) in \(\mathbb{R}^d\).