Let's analyze each statement to determine which one is true:

a. A convex set can have infinitely many extreme points.

• This is true. For example, a convex set like a circle has infinitely many extreme points (the points on the perimeter).

b. A linear programming problem can have infinitely many extreme points.

• This is generally false. A linear programming problem is typically defined on a convex polytope with a finite number of vertices (extreme points).

c. A linear programming problem can have exactly two different optimal solutions.

• This is false. If a linear programming problem has multiple optimal solutions, they form a line segment (infinite solutions) rather than just two distinct points.

d. A linear programming problem cannot have non-basic optimal solutions.

• This is false. Linear programming problems can have non-basic optimal solutions along the edges of the feasible region.

Conclusion:

The true statement is a. A convex set can have infinitely many extreme points.

Would you like further details or have any questions? Here are some related questions you might consider:

1. What are examples of convex sets with infinitely many extreme points?
2. How do extreme points relate to the feasibility of a linear programming problem?
3. Can a linear programming problem have multiple optimal solutions without being unbounded?
4. What is the significance of basic and non-basic solutions in linear programming?
5. How is the convexity of a set determined mathematically?

Tip: Always visualize convex sets and their extreme points using graphs to better understand their properties!