second order stochastic dominance for convex functions

Hi,

To gain better understanding of second-order stochastic dominance I am trying to replicate some basic proofs. The proof of `standard' SSD (relating to a concave utility function) is okay but that for convex utility functions seems to be a lot harder.
Recall that SSD holds iff: for any y.

For the convex version of SSD, often referred to as Descending SSD, the condition above becomes: for any x.

I have found a paper (Levy & Wiener (1998) that gives two hints but does not provide the proof itself. The first hint is that if u(x) is convex <-> -u(-x) is concave. I have managed to show that SSD holds when -u(-x) is concave (the pitfall there is to not mess up any of the minus-signs for the first and second derivative...), but the second hint remains unclear to me. They suggest that x DSSD y <-> -y SSD -x, so that plugging in -y SSD -x into the first condition above, this immediately leads to x DSSD y.

Can anyone show me some intermediate steps of the derivation? I have the gut feeling that a survival function should show up somewhere (i.e. (1-F(x)) ) but I cannot seem to understand where or when in the derivation.

It is stating and proving that the expected utility is larger for every weakly increasing concave utility function;
and you want to show the given condition is equivalent to this, in the descending SSD version (with convex one)?