Vol. 304, No. 2, 2020

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Centers of disks in Riemannian manifolds

Igor Belegradek and Mohammad Ghomi

Vol. 304 (2020), No. 2, 401–418

We prove the existence of a center, or continuous selection of a point, in the relative interior of C1 embedded k-disks in Riemannian n-manifolds. If k 3 the center can be made equivariant with respect to the isometries of the manifold, and under mild assumptions the same holds for k = 4 = n. By contrast, for every n k 6 there are examples where an equivariant center does not exist. The center can be chosen to agree with any of the classical centers defined on the set of convex compacta in the Euclidean space.

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continuous selection, equivariant, proper actions, actions on disks
Mathematical Subject Classification 2010
Primary: 53C40, 54C65
Secondary: 52A20, 57S25
Received: 21 May 2018
Revised: 1 July 2019
Accepted: 12 July 2019
Published: 12 February 2020
Igor Belegradek
School of Mathematics
Georgia Institute of Technology
Atlanta, GA
United States
Mohammad Ghomi
School of Mathematics
Georgia Institute of Technology
Atlanta, GA
United States