Journal of Formalized Mathematics
Volume 14, 2002
University of Bialystok
Copyright (c) 2002 Association of Mizar Users

## 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

The terminology and notation used in this paper have been introduced in the following articles [12] [11] [17] [9] [13] [3] [1] [8] [4] [2] [15] [14] [16] [5] [10] [6] [7]

#### Contents (PDF format)

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

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