 
Summary: Mathematical Programming I:
Chapter 4 and Chapter 5
Sheets for the course, version: 22052011
Georg Still
The script is part of the book:
Faigle/Kern/Still, Algorithmic principles of Mathematical
Programming.
on: http://wwwhome.math.utwente.nl/stillgj/priv/
Math. Prog. Ch.4,5 p 1
4. Convex sets, convex functions
4.1 Convex sets
Recall the definitions of closed, compact sets P Rn.
Def. A set P Rn is called convex if
x, y P, [0, 1] x + (y  x) P
Ex 4.1 Any intersection of closed convex sets is a closed
convex set.
Def. Let H be a hyperplane H = {x  aT x = } (with some
0 = a Rn) and let P Rn, y / P. H is called a separating
hyperplane wrt. P and y if:
aT
