Lecture Notes in Convexity
Instructor: Renato D. C. Monteiro
November 27, 2010
Chapter 1
Convex sets
1.1
Notation
In this section, we introduce some global notation and terminology that will be used
throughout our presentation.
We write V W to indicate th
ISyE 7682
Fall 2010 1st Homework
Problem 0.1 Assume that L is a subspace and A Rn is an ane manifold. Show that:
(a) for any x0 n , the set L + x0 is an ane manifold;
(b) for any x0 A, the set A x0 is a subspace which does not depend on x0 .
(c) A is a su
ISyE 7682
Fall 2010 2nd Homework
Problem 0.1 Let Ci ni be a convex set for i = 1, 2. Show that
a C1 a C2 = a (C1 C2 ),
ri C1 ri C2 = ri (C1 C2 ).
Problem 0.2 For a nonempty closed convex C n , show that C = cfw_d n : C + d C .
Problem 0.3 Let = C n be a c
ISyE 7682
Fall 2010 3rd Homework
Problem 0.1 For a set S n , show the following equivalences:
a) S is closed if, and only if, IS is a lower semi-continuous function;
b) S is convex if, and only if, IS Conv(n );
c) S is a nonempty closed convex set if, and
ISyE 7682
FALL 2010 1st Homework
Problem 0.1 Assume that L is a subspace and A Rn is an ane manifold. Show that:
(a) for any x0 n , the set L + x0 is an ane manifold;
(b) for any x0 A, the set A x0 is a subspace which does not depend on x0 .
(c) A is a su
ISyE 7682
FALL 2010 2nd Homework
Problem 0.1 Let Ci ni be a convex set for i = 1, 2. Show that
a C1 a C2 = a (C1 C2 ),
ri C1 ri C2 = ri (C1 C2 ).
(1)
(2)
Solution. We rst show (1). It is easy to see that if A1 n1 and A2 n are two ane
2
sets then A1 A2 n1
ISyE 7682
Fall 2010 3rd Homework
Problem 0.1 For a set S n , show the following equivalences:
a) S is closed if, and only if, IS is a lower semi-continuous function;
b) S is convex if, and only if, IS Conv(n );
c) S is a nonempty closed convex set if, and
ISYE 7682
Convexity (Fall 2010)
Instructor: Renato D.C. Monteiro.
Oce: Groseclose Bldg 424.
Phone #: (404)894-1450.
E-mail: [email protected]
Oce hours: MW 4:00-5:00
Time and Place: MW 5-6:30 on IC 119.
Description: In this course I will give an in