Minimal Lipschitz and ∞-harmonic extensions of vector-valued functions on finite graphs

This paper deals with extensions of vector-valued functions on finite graphs fulfilling distinguished minimality properties. We show that so-called $\mathrm{lex}$ and $L\mbox{-}\mathrm{lex}$ minimal extensions are actually the same and call them minimal Lipschitz extensions. Then, we prove that the solution of the graph $p$-Laplacians converge to these extensions as $p\to \infty$. Furthermore, we examine the relation between minimal Lipschitz extensions and iterated weighted midrange filters and address their connection to $\infty$-Laplacians for scalar-valued functions. A convergence proof for an iterative algorithm proposed by Elmoataz et al. (2014) for finding the zero of the $\infty$-Laplacian is given. Finally, we present applications in image inpainting.

Lipschitz Extensions to Finitely Many Points

We consider Lipschitz maps with values in quasi-metric spaces and extend such maps to finitely many points. We prove that in this context every 1-Lipschitz map admits an extension such that its Lipschitz constant is bounded from above by the number of added points plus one. Moreover, we prove that if the source space is a Hilbert space and the target space is a Banach space, then there exists an extension such that its Lipschitz constant is bounded from above by the square root of the total of added points plus one. We discuss applications to metric transforms.

Optimal Lipschitz extensions

The classical Lipschitz extension problem in concerned for conditions on a pair of metric spaces (X,dX) and (Y,dY) such that for all Ω ⊂ X and for all Lipschitz function and for all Lipschitz function f : Ω → Y, then there is a function g : X → Y that extends f and has the same Lipschitz constant as f . In this paper we discuss some results and open questions related to that issue.