@MISC{CONVEX1.MIZ,
  AUTHOR = {Endou, Noboru and Mitsuishi, Takashi and Shidama, Yasunari},
  TITLE = {Convex Sets and Convex Combinations},
  SECTION1 = {Convex Sets},
  SECTION2 = {Convex Combinations},
  SECTION3 = {Convex Hull},
  SECTION4 = {Miscellaneous},
  DAY = {05},
  MONTH = {November},
  YEAR = {2002},
  ADDRESS1 = {Gifu National College of Technology},
  ADDRESS2 = {Miyagi University},
  ADDRESS3 = {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.}}

