We pursue two goals in this article. As our first goal, we construct a family M G of Gibbs like measures on the set of piecewise linear convex functions g : R 2 → R. It turns out that there is a one-to-one correspondence between the gradient of such convex functions and Laguerre tessellations. Each cell in a Laguerre tessellation is a convex polygon that is marked by a vector ρ ∈ R 2 . Each measure ν f ∈ M G in our family is uniquely characterized by a kernel f (x, ρ − , ρ + ), which represents the rate at which a line separating two cells associated with marks ρ − and ρ + passes through x. To construct our measures, we give a precise recipe for the law of the restriction of our tessellation to a box. This recipe involves a boundary condition, and a dynamical description of our random tessellation inside the box. As we enlarge the box, the consistency of these random tessellations requires that the kernel satisfies a suitable kinetic like PDE. Our interest in the family M G stems from its close connection to Hamilton-Jacobi PDEs of the form u t = H(u x ), with u : Γ → R, for a convex set Γ ⊂ R 2 × [0, ∞), and H : R 2 → R a strictly convex function. As our second goal, we study the invariance of the set M G with respect to the dynamics of such Hamilton-Jacobi PDEs. In particular we conjecture the invariance of a suitable subfamily M G of M G . More precisely, we expect that if the initial slope u x (•, 0) is selected according to a measure ν f ∈ M G , then at a later time the law of u x (•, t) is given by a measure ν Θt(f ) ∈ M G , for a suitable kernel Θ t (f ). As we vary t, the kernel Θ t (f ) must satisfy a suitable kinetic equation. We remark that the function u is also piecewise linear convex function in (x, t), and its law is an example of a Gibbs-like measure on the set of Laguerre tessellations of certain convex subsets of R 3 .[R2] F. Rezakhanlou, Kinetic description of scalar conservation laws with Markovian data, in preparation.