Convex domains and linear combinations of continuous functions

Convex Domains and Linear Combinations of Continuous Functions

Convex Sets and Convex Combinations on Complex Linear Spaces

Let V be a non empty zero structure. An element of Cthe carrier of V is said to be a C-linear combination of V if: (Def. 1) There exists a finite subset T of V such that for every element v of V such that v / ∈ T holds it(v) = 0. Let V be a non empty additive loop structure and let L be an element of Cthe carrier of V . The support of L yielding a subset of V is defined by: (Def. 2) The support...

Convex Defining Functions for Convex Domains

We give three proofs of the fact that a smoothly bounded, convex domain in R n has defining functions whose Hessians are non-negative definite in a neighborhood of the boundary of the domain.

Identifying Linear Combinations of Ridge Functions

This paper is about an inverse problem. We assume we are given a function f(x) which is some sum of ridge functions of the form ∑m i=1 gi(a i · x) and we just know an upper bound on m. We seek to identify the functions gi and also the directions a i from such limited information. Several ways to solve this nonlinear problem are discussed in this work. §

Convex Sets and Convex Combinations

Convexity is one of the most important concepts in a study of analysis. Especially, it has been applied around the optimization problem widely. Our purpose is to define the concept of convexity of a set on Mizar, and to develop the generalities of convex analysis. The construction of this article is as follows: Convexity of the set is defined in the section 1. The section 2 gives the definition...