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Convex Analysis in Hadamard Spaces

Presented by: 
Gabriele Steidl University of Kaiserslautern
Wednesday 6th September 2017 - 16:10 to 17:00
INI Seminar Room 1
joint work with M. Bacak, R. Bergmann, M. Montag and J. Persch

The aim of the talk is two-fold:

1. A well known result of H. Attouch states that the Mosco convergence of a sequence of proper convex lower semicontinuous functions defined on a Hilbert space is equivalent to the pointwise convergence of the associated Moreau envelopes. In the present paper we generalize this result to Hadamard spaces. More precisely, while it has already been known that the Mosco convergence of a sequence of convex lower semicontinuous functions on a Hadamard space implies the pointwise convergence of the corresponding Moreau envelopes, the converse implication was an open question. We now fill this gap.  Our result has several consequences. It implies, for instance, the equivalence of the Mosco and Frolik-Wijsman convergences of convex sets. As another application, we show that there exists a~complete metric on the cone of proper convex lower semicontinuous functions on a separable Hadamard space such that a~sequence of functions converges in this metric if and only if it converges in the sense of Mosco.

2. We extend the parallel Douglas-Rachford algorithm  to the manifold-valued setting.
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons