The purpose of the present paper is to prove, solely using Convex (and Nonsmooth) analysis techniques, that Hopf-Lax formulae provide explicit solutions for Hamilton-Jacobi equations with merely lower semicontinuous initial data. The substance of these results appears in a paper by Alvarez, Barron and Ishii (1999) but the proofs are fundamentally different (we do not use the comparison principle) and a distinct notion of discontinuous solutions is used. Moreover we give a maximum principle for the Lax function. This approach permits us to fully understand the role of the convexity of the data.