Inf-Convolution and Regularization of Convex Functions on Riemannian Manifolds of Nonpositive Curvature

Abstract

We show how an operation of inf-convolution can be used to approximate convex functions with C 1 smooth convex functions on Riemannian manifolds with nonpositive curvature (in a manner that not only is explicit but also preserves some other properties of the original functions, such as ordering, symmetries, infima and sets of minimizers), and we give some applications.

Global and fine approximation of convex functions

“…A similar statement holds for C 1 -fine approximation. By combining this observation with [2,Corollary 4.4], we also deduce the following: if M is a complete finite-dimensional Riemannian manifold with sectional curvature K 0, then every function in P(M ) can be approximated by C 1 convex functions in the C 0 -fine topology. The condition that f belong to P(M ) cannot be removed in general, as we already know by considering the case when M = R n , or when M is one of the manifolds constructed in [20].…”

“…Property (iv) is obvious. To see that the reverse inequality holds on B, take x ∈ B and a subdifferential ζ ∈ D − f (x) (we refer to [3], [2] for the definitions and some properties of the Fréchet subdifferential and inf convolution on Riemannian manifolds). We have ζ x ≤ L because f is L-Lipschitz on B.…”

“…See [24] for a survey on the inf convolution operation in Banach spaces, and [2] for the Cartan-Hadamard case.…”

“…For instance [12], every two-dimensional manifold which admits a global convex function that is locally nonconstant must be diffeomorphic to the plane, the cylinder, or the open Möbius strip. 2 We warn the reader that, in Greene and Wu's papers, what we have just defined as strong convexity is called strict convexity. We have changed their terminology since we will be mainly concerned with the case M = R d , where one traditionally defines a strictly convex function as a function f satisfying f ((1 − t)x + ty) < (1 − t)f (x) + tf (y) if 0 < t < 1. very strong condition: for instance, the function f (x) = x 4 is strictly convex, but not strongly convex on any neighbourhood of 0.…”

Regularization by sup–inf convolutions on Riemannian manifolds: An extension of Lasry–Lions theorem to manifolds of bounded curvature

“…When M = R n or a Hilbert space, it is a well known fact (and easy to prove) that the operation f → f λ preserves global or local Lipschitz and convexity properties of f . In the Riemannian setting one has to impose curvature restrictions on M in order to obtain similar results, see [4], and the proofs are somewhat subtler.…”