2. Some good solutions here, but many students had difficulty with this question. The factors were:
- Uncertain about what needed to be shown:
- a set S is convex if x1,x2 \in S and t \in [0,1] implies t*x1+(1-t)*x2 \in S
- a set S is affine if x1,x2 \in S and t \in R implies t*x1+(1-t)*x2 \in S
- Had difficulty communicating clearly (algebraically) what needed to be shown.
- Claiming something to be true, but not proving it.
1.5: Most students did not know how to deal with the points (x,t) \in epi(f) to show convexity. Many thought the elements of epi(f) were simply x.
1.7: No problems here.
1.8: 1. Most students realized the extreme points were e1,...,en,
but not everyone provided a motivation for their answer.
2. Some people just referred to the theorem that every compact convex set
is the convex hull of its extreme points, but it is much more straightforward
to just use the definition of conv(S).
3. Most students didn't conclude that aff(C) = {x | \sum(xi, i=1..n) = 1}
4. No problems here.
1.13: The biggest problem here was that students didn't know how to write the dual cone of S_n. For example, some wrote the matrices as vectors or were unsure about how to write the matrix inner-product. Another issue was that some people only showed that S_n \subset (S_n)^*, but not the other direction.
1.18: No problems here other than sometimes choosing a function h which was not convex.