Abstract
A Krasnosel’skii-type theorem for compact sets that are starshaped via staircase paths may be extended to compact sets that are starshaped via orthogonally convex paths: Let S be a nonempty compact planar set having connected complement. If every two points of S are visible via orthogonally convex paths from a common point of S, then S is starshaped via orthogonally convex paths. Moreover, the associated kernel Ker S has the expected property that every two of its points are joined in Ker S by an orthogonally convex path. If S is an arbitrary nonempty planar set that is starshaped via orthogonally convex paths, then for each component C of Ker S, every two of points of C are joined in C by an orthogonally convex path.
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Communicated by Imre Bárány
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Breen, M. A Krasnosel’skii-type result for planar sets starshaped via orthogonally convex paths. Period Math Hung 55, 169–176 (2007). https://doi.org/10.1007/s10998-007-4169-8
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DOI: https://doi.org/10.1007/s10998-007-4169-8