Convex Sets and Convex Combinations

Noboru Endou

Gifu National College of Technology

Takashi Mitsuishi

Miyagi University

Yasunari Shidama

Shinshu University, Nagano

Summary.

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 of convex combination which is a kind of the linear combination and related theorems are proved there. In section 3, we define the convex hull which is an intersection of all convex sets including a given set. The last section is some theorems which are necessary to compose this article.

MML Identifier: CONVEX1

Contents (PDF format)

1. Convex Sets
2. Convex Combinations
3. Convex Hull
4. Miscellaneous

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