Related papers: Absolutely minimal Lipschitz extension of tree-val…

Absolutely minimal Lipschitz extensions (AMLEs) are known to exist in many infinite metric settings, but the finite case is less settled. In metric spaces with at most four points, every function on a nonempty subset admits an AMLE in the…

We prove an asymptotic Lipschitz estimate for value functions of tug-of-war games with varying probabilities defined in $\Omega\subset \mathbb R^n$. The method of the proof is based on a game-theoretic idea to estimate the value of a…

We define a random step size tug-of-war game, and show that the gradient of a value function exists almost everywhere. We also prove that the gradients of value functions are uniformly bounded and converge weakly to the gradient of the…

Let (X,d) be a metric space and $ \alpha > 0 $. In this paper, we study extensions of some complex-valued Lipschitz functions, from some special subset $ X_0 $ to X. These extensions are with no-increasing Lipschitz number or the smallest…

We consider a real-valued function $f$ defined on the set of infinite branches $X$ of a countably branching pruned tree $T$. The function $f$ is said to be a \textit{limsup function} if there is a function $u \colon T \to \mathbb{R}$ such…

In this paper we consider a wide class of generalized Lipschitz extension problems and the corresponding problem of finding absolutely minimal Lipschitz extensions. We prove that if a minimal Lipschitz extension exists, then under certain…

The classical McShane-Whitney extension theorem for Lipschitz functions is refined by showing that for a closed subset of the domain, it remains valid for any interval of the real line. This result is also extended to the setting of locally…

In cooperative game theory, the primary focus is the equitable allocation of payoffs or costs among agents. However, in the practical applications of cooperative games, accurately representing games is challenging. In such cases, using an…

We study the variational structure of the biased infinity Laplacian by introducing a notion of the $\beta$\textit{-Exponential Absolute Minimizing Extension} ($\beta$--AM) on arbitrary length space, which absolutely minimizing the…

We study the problem of extending an order-preserving real-valued Lipschitz map defined on a subset of a partially ordered metric space without increasing its Lipschitz constant and preserving its monotonicity. We show that a certain type…

A mapping $f:X\to Y$ between metric spaces is called \emph{little Lipschitz} if the quantity $$ \operatorname{lip}(f(x)=\liminf_{r\to0}\frac{\operatorname{diam} f(B(x,r))}{r} $$ is finite for every $x\in X$. We prove that if a compact (or,…

The tent map is an elementary example of an interval map possessing many interesting properties, such as dense periodicity, exactness, Lipschitzness and a kind of length-expansiveness. It is often used in constructions of dynamical systems…

Sparse goal-conditioned planning with few cost-to-go labels can be viewed as a graph-PDE Dirichlet extension problem: extend sparse labels on a goal-dependent boundary to unlabelled graph vertices so that greedy rollouts reach the goal. We…

We study a new bi-Lipschitz invariant \lambda(M) of a metric space M; its finiteness means that Lipschitz functions on an arbitrary subset of M can be linearly extended to functions on M whose Lipschitz constants are enlarged by a factor…

A multiple-valued function $f:X\to {\bf Q}_Q(Y)$ is essentially a rule assigning $Q$ unordered and non necessarily distinct elements of $Y$ to each element of $X$. We study the Lipschitz extension problem in this context by using two…

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…

We give a simple quantitative condition, involving the "mapping content" of Azzam--Schul, that implies that a Lipschitz map from a Euclidean space to a metric space must be close to factoring through a tree. Using results of Azzam--Schul…

In this paper, we study games with continuous action spaces and non-linear payoff functions. Our key insight is that Lipschitz continuity of the payoff function allows us to provide algorithms for finding approximate equilibria in these…

We study the relations between the Lipschitz constant of $1$-field and the Lipschitz constant of the gradient canonically associated with this $1$-field. Moreover, we produce two explicit formulas that make up Minimal Lipschitz extensions…

We prove that any correspondence (multi-function) mapping a metric space into a Banach space that satisfies a certain pointwise Lipschitz condition, always has a continuous selection that is pointwise Lipschitz on a dense set of its domain.…