It is well known that the log-concavity of a distribution is preserved under affine transformation (see e.g. Dharmadhikari and Joag-Dev [1988], Lemma 2.1).
Knows someone a reference to a theorem that would say that this is true also for strict log-concave case? My guess is that the claim is true, but cannot find a proof.
(Note that the above reference uses log-concave measures to proof the claim, thus the implication is not clear.)