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It is shown that f is convex if and only if, for all x,y ∈ D, there exists α = α(x,y) ∈ (0, 1) such that f (αx + (1 − α)y) ≤ αf (x) + (1 − α)f (y). for all x,y ∈ D. It is clear that convex functions are α-convex.
Let's first recall the definition of a convex function. Definition 1. A function f : Rn → R is convex if its domain is a convex set and for all x, y in ...
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Jul 26, 2022 · Let f:I→R be a function, if f is convex if then (B1),(B2) and (B3) holds. furthermore the non differentiable points of f are at most countable.
The main purpose of this paper is to characterize convex functions. A characterization of a locally convex set is also given.
We will first give an important characterization of convex function. To so, we need to characterize multivariate functions via their Taylor expansion. 2.1 ...
Sep 25, 2017 · Abstract:Let D be a convex subset of a real vector space. It is shown that a radially lower semicontinuous function f: D\to \mathbf{R}\cup ...
In mathematics, a real-valued function is called convex if the line segment between any two distinct points on the graph of the function lies above or on the ...
Our aim in this section is to show that an arbitrary function from. C(K,R) can be approximated in the supremum norm by a sum (with non- negative coefficients) ...
Abstract. If C ⊆ Rn be a nonempty convex set, then f: C → R is convex function if and only if it is a quasiconvex function on C and there exists some α ∈ (0, 1) ...
We give a characterization of convex functions in terms of difference among values of a function. As an application, we propose an estimation of operator ...