Show if the set is convex or not!
I know that to show a set is convex you can either use the definition or show that the set can be obtained from known convex sets under operations that preserve convexity.
Convex definition: x1*Theta + (1 - Theta)*x2 ∈ S2
Ops that preserve convex sets (that i know of): Intersection, affine functions, perspective functions, linear functionals.
This formula may come in handy:
The Attempt at a Solution
I was able to rewrite the set as:
I have no idea if that will help me and I also have no idea where to go from here!!! Using the definition I end up with sums in the log function which leads me no where. I have a feeling this set is not convex but once again I am not sure.
Please any help will be greatly appreciated as I have been thinking about it for a while.
PS: I am able to prove simple things like the convexity of a norm ball or polyhedra but I am not able to do this one.
EDIT: I have scanned a better attempt at this problem, however I am still not able to do it. I feel defeated, it feels bad : (