Given two functions, $f$ which is convex and $g$ which is strictly convex, is the difference $f-g$ convex?
My impulse is to say no, since $-g$ should be concave, but I'm trying to show this explicitly.
I've been trying to use the definition of convexity/strict convexity to get somewhere on this: $$f(\lambda x+(1-\lambda )y) \le \lambda f(x) + (1-\lambda )f(y)$$ $$g(\lambda x+(1-\lambda )y) < \lambda g(x) + (1-\lambda )g(y)$$
But to see anything useful I'd need to subtract the second equation from the first one. Can I do that? We'd then end up with a weird combination of less than and greater than statements, which feels wrong.
Any suggestions here? Is there a better way to get a result? Thank you!
Just pick a counter example, let $f=0$ and $g=x^4$,
then $f-g=-x^4$ and it is not convex.