Abstract
We show that if \((X,d)\) is a metric space which admits a consistent convex geodesic bicombing, then we can construct a conical bicombing on \(CB(X)\), the hyperspace of nonempty, closed, bounded, and convex subsets of \(X\) (with the Hausdorff metric). If \(X\) is a normed space or an \(\mathbb {R}\)-tree, this same method produces a consistent convex bicombing on \(CB(X)\). We follow this by examining a geodesic bicombing on the nonempty compact subsets of \(X\), assuming \(X\) is a proper metric space.
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Notes
Other authors may call this \(\sigma \)-convex, as it does in fact depend on the bicombing \(\sigma \).
Every complete and locally compact space with intrinsic metric is a proper geodesic space.
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Acknowledgements
The author would like to thank Peter Oberly, Joel H.Shapiro, and J.J.P. Veerman for many helpful comments.
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Fox, L.S. Geodesic bicombings on some hyperspaces. J. Geom. 113, 28 (2022). https://doi.org/10.1007/s00022-022-00642-6
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DOI: https://doi.org/10.1007/s00022-022-00642-6