A Q about S.Boyd's CVX book p. 107

Question

A Q about the following: (Come from S.Boyd's note)

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Note 1: y is a r.v with log-concave pdf "p(y)".

My Qs are:

1. What is the f(x) in the proof? It is marginal pdf or cdf or expected value? It seems like an expected value?

   

2. Why is 'g(u)' a log-concave function?

Thanks!

Answer 1

1) $f(x)$ is what the question says it is: the probability that $x + y \in C$. Since $g(x+y)$ is $1$ when $x+y \in C$ and $0$ otherwise, this is indeed the expected value of $g(x+y)$.

2) $g$ is log-concave because $C$ is a convex set. This follows easily from the definitions of log-concave function and convex set.