Vol. 304, No. 2, 2020

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Centers of disks in Riemannian manifolds

Vol. 304 (2020), No. 2, 401–418
Abstract

We prove the existence of a center, or continuous selection of a point, in the relative interior of ${C}^{1}$ embedded $k$-disks in Riemannian $n$-manifolds. If $k\le 3$ 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 $n\ge k\ge 6$ 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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Keywords
continuous selection, equivariant, proper actions, actions on disks
Mathematical Subject Classification 2010
Primary: 53C40, 54C65
Secondary: 52A20, 57S25
Milestones
Received: 21 May 2018
Revised: 1 July 2019
Accepted: 12 July 2019
Published: 12 February 2020
Authors
 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