Related papers: Convex and quasiconvex functions in metric graphs

In this short note we consider an unconventional overdetermined problem for the torsion function: let $n\geq 2$ and $\Omega$ be a bounded open set in $\mathbb{R}^n$ whose torsion function $u$ (i.e. the solution to $\Delta u=-1$ in $\Omega$,…

We quickly review and make some comments on the concept of convexity in metric spaces due to Takahashi. Then we introduce a concept of convex structure based convexity to functions on these spaces and refer to it as $W-$convexity.…

We show that if a sequence of dense graphs has the property that for every fixed graph F, the density of copies of F in these graphs tends to a limit, then there is a natural ``limit object'', namely a symmetric measurable 2-variable…

Two criteria for the robust quasiconvexity of lower semicontinuous functions are established in terms of Fr\'echet subdifferentials in Asplund spaces.

We consider Novikov problem of the classification of level curves of quasiperiodic functions on the plane and its connection with the conductivity of two-dimensional electron gas in the presence of both orthogonal magnetic field and the…

Let $\mathcal{G} = \{G_1 = (V, E_1), \dots, G_m = (V, E_m)\}$ be a collection of $m$ graphs defined on a common set of vertices $V$ but with different edge sets $E_1, \dots, E_m$. Informally, a function $f :V \rightarrow \mathbb{R}$ is…

Given a graph with non-negative edge weights, there are various ways to interpret the edge weights and induce a metric on the vertices of the graph. A few examples are shortest-path, when interpreting the weights as lengths; resistance…

We consider the following two algorithmic problems: given a graph $G$ and a subgraph $H\subseteq G$, decide whether $H$ is an isometric or a geodesically convex subgraph of $G$. It is relatively easy to see that the problems can be solved…

A matching cut in a graph G is an edge cut of G that is also a matching. This short survey gives an overview of old and new results and open problems for Maximum Matching Cut, which is to determine the size of a largest matching cut in a…

In this paper, we reformulate the Bakry-\'Emery curvature on a weighted graph in terms of the smallest eigenvalue of a rank one perturbation of the so-called curvature matrix using Schur complement. This new viewpoint allows us to show…

We construct a H\"older continuous function on the unit interval which coincides in uncountably (in fact continuum) many points with every function of total variation smaller than 1 passing through the origin. We say that a function with…

A transitive graph is 2-dimensional if it can be represented as the intersection of two linear orders. Such representations make answering of reachability queries trivial, and allow many problems that are NP-hard on arbitrary graphs to be…

In this paper, we present a new metric distance for comparing two large graphs to find similarities and differences between them based on one of the most important graph structural properties, which is Node Adjacency Information, for all…

We prove that if f : R^N --> R is quasiconvex and U is open in the density topology of R^N, then sup_U f = ess sup_U f, while inf_U f = ess inf_U f if and only if the equality holds when U = R^N. The first (second) property is typical of…

We study a specific convex maximization problem in the space of continuous functions defined on a semi-infinite interval. An unexplained connection to the discrete version of this problem is investigated.

If $u : \Omega\subset \mathbb{R}^d \to {\rm X}$ is a harmonic map valued in a metric space ${\rm X}$ and ${\sf E} : {\rm X} \to \mathbb{R}$ is a convex function, in the sense that it generates an ${\rm EVI}_0$-gradient flow, we prove that…

Subdifferentials (in the sense of convex analysis) of matrix-valued functions defined on $\mathbb{R}^d$ that are convex with respect to the L\"{o}wner partial order can have a complicated structure and might be very difficult to compute…

Let $G$ be a connected graph. Given an ordered set $W = \{w_1, w_2,\dots w_k\}\subseteq V(G)$ and a vertex $u\in V(G)$, the representation of $u$ with respect to $W$ is the ordered $k$-tuple $(d(u,w_1), d(u,w_2),\dots,$ $d(u,w_k))$, where…

In this paper, we unify and improve existing results on characterizing strict and almost stricty convex functions via subdifferential mapping, Moreau envelope, and proximal mappings. In particular, it is shown that if a convex function is…

Submodular continuous functions are a category of (generally) non-convex/non-concave functions with a wide spectrum of applications. We characterize these functions and demonstrate that they can be maximized efficiently with approximation…