  The objective of this paper is to find some inequalities satisfied by
periodical solutions of multi-time Hamilton systems, when the Hamiltonian is
convex. To our knowledge, this subject of first-order field theory is still
open.
  Section 1 recall well-known facts regarding the equivalence between
Euler-Lagrange equations and Hamilton equations and analyses the action that
produces multi-time Hamilton equations, emphasizing the role of the
polysymplectic structure. Section 2 extends two inequalities of [21] from a
cube to parallelipiped and proves two inequalityes concerning multiple
periodical solutions of multi-time Hamilton equations.
