We prove that Thompson’s group F (n) is not minimally almost convex with respect to the standard finite generating set. A group G with Cayley graph Γ is not minimally almost convex if for arbitrarily large values of m there exist elements g, h ∈ Bm such that dΓ(g, h) = 2 and dBm (g, h) = 2m. (Here Bm is the ball of radius m centered at the identity.) We use tree-pair diagrams to represent eleme...

The author writes in the preface: “Discrete Convex Analysis is aimed at establishing a novel theoretical framework for solvable discrete optimization problems by means of a combination of the ideas in continuous optimization and combinatorial optimization.” Thus the reader may conclude that the book presents a new theory (the name “discrete convex analysis” was, apparently, coined by the author...

We prove that Thompson’s group F (n) is not minimally almost convex with respect to the standard finite generating set. A group G with Cayley graph Γ is not minimally almost convex if for arbitrarily large values of m there exist elements g, h ∈ Bm such that dΓ(g, h) = 2 and dBm (g, h) = 2m. (Here Bm is the ball of radius m centered at the identity.) We use tree-pair diagrams to represent eleme...

Definition 1.1. Let C be a subset of R. We say C is convex if αx+ (1− α)y ∈ C, ∀x, y ∈ C, ∀α ∈ [0, 1]. Definition 1.2. Let C be a convex subset of R. A function f : C 7→ R is called convex if f(αx+ (1− α)y) ≤ αf(x) + (1− α)f(y), ∀x, y ∈ C, ∀α ∈ [0, 1]. The function f is called concave if −f is convex. The function f is called strictly convex if the above inequality is strict for all x, y ∈ C wi...

A number of learning problems can be cast as an Online Convex Game: on each round, a learner makes a prediction x from a convex set, the environment plays a loss function f , and the learner’s long-term goal is to minimize regret. Algorithms have been proposed by Zinkevich, when f is assumed to be convex, and Hazan et al., when f is assumed to be strongly convex, that have provably low regret. ...

A number of learning problems can be cast as an Online Convex Game: on each round, a learner makes a prediction x from a convex set, the environment plays a loss function f , and the learner’s long-term goal is to minimize regret. Algorithms have been proposed by Zinkevich, when f is assumed to be convex, and Hazan et al., when f is assumed to be strongly convex, that have provably low regret. ...

‎In this paper‎, ‎we generalize the proximal point algorithm to complete CAT(0) spaces and show‎ ‎that the sequence generated by the proximal point algorithm‎ $w$-converges to a zero of the maximal‎ ‎monotone operator‎. ‎Also‎, ‎we prove that if $f‎: ‎Xrightarrow‎ ‎]-infty‎, +‎infty]$ is a proper‎, ‎convex and lower semicontinuous‎ ‎function on the complete CAT(0) space $X$‎, ‎then the proximal...

It follows from Browder (Summa Bras Math 4:183–191, 1960) that for every continuous function $$F : (X \times Y) \rightarrow Y$$ , where X is the unit interval and Y a nonempty, convex, compact subset of locally convex linear vector space, set fixed points F, defined by $$C_F := \{ (x,y) \in :F(x,y)=y\}$$ has connected component whose projection to first coordinate X. We extend Browder’s result ...

In this paper, strongly (α ,T ) -convex functions, i.e., functions f : D → R satisfying the functional inequality f (tx+(1− t)y) t f (x)+(1− t) f (y)− tα(1− t)(x− y)− (1− t)αt(y− x) for x,y ∈ D and t ∈ T ∩ [0,1] are investigated. Here D is a convex set in a linear space, α is a nonnegative function on D−D , and T ⊆ R is a nonempty set. The main results provide various characterizations of stron...