On Discrete Hessian Matrix and Convex Extensibility

For functions defined on integer lattice points, discrete versions of the Hessian matrix have been considered in various contexts. In discrete convex analysis, for example, certain combinatorial properties of the discrete Hessian matrices are known to characterize M-convex and L-convex functions, which can be extended to convex functions in real variables. The relationship between convex extensibility and discrete Hessian matrices is not fully understood in general, and unfortunately, some vague or imprecise statements have been made in the literature. This note points out that the positive semidefiniteness of the discrete Hessian matrix does not imply nor is implied by convex extensibility of discrete functions.