Lipschitz Functions on Expanders are Typically Flat

Lectures on Lipschitz Analysis

(1.1) |f(a)− f(b)| ≤ L |a− b| for every pair of points a, b ∈ A. We also say that a function is Lipschitz if it is L-Lipschitz for some L. The Lipschitz condition as given in (1.1) is a purely metric condition; it makes sense for functions from one metric space to another. In these lectures, we concentrate on the theory of Lipschitz functions in Euclidean spaces. In Section 2, we study extensio...

Grounded Lipschitz functions on trees are typically flat

A grounded M -Lipschitz function on a rooted d-ary tree is an integer-valued map on the vertices that changes by at most M along edges and attains the value zero on the leaves. We study the behavior of such functions, specifically, their typical value at the root v0 of the tree. We prove that the probability that the value of a uniformly chosen random function at v0 is more than M + t is doubly...

An effective optimization algorithm for locally nonconvex Lipschitz functions based on mollifier subgradients

POINT DERIVATIONS ON BANACH ALGEBRAS OF α-LIPSCHITZ VECTOR-VALUED OPERATORS

The Lipschitz function algebras were first defined in the 1960s by some mathematicians, including Schubert. Initially, the Lipschitz real-value and complex-value functions are defined and quantitative properties of these algebras are investigated. Over time these algebras have been studied and generalized by many mathematicians such as Cao, Zhang, Xu, Weaver, and others. Let  be a non-emp...

Compact composition operators on real Banach spaces of complex-valued bounded Lipschitz functions

We characterize compact composition operators on real Banach spaces of complex-valued bounded Lipschitz functions on metric spaces, not necessarily compact, with Lipschitz involutions and determine their spectra.