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In this work we provide a general framework to construct approximations of the space of convex functions on a bounded domain that satisfies a Lipschitz bound. Our approximating space is a finite-dimensional polyhedron, that is a subset of a finite-dimensional functional space that satisfies a finite number of linear constraints. Our main theoretical contribution of this article is a bound on the (Hausdorff) distance between the approximating polyhedron and the admissible set of convex functions.
This type of discretization is well suited to the numerical resolution of problems of calculus of variations under convexity constraints. For instance, we show how to compute the projection onto the discretized space of convex functions in dimension by combining a proximal algorithm and an efficient projection operator on the space of 1D discrete convex functions. See the above illustration.
Finally, we note that the discretization of the space of convex functions we propose can be generalized to other spaces of functions satisfying similar constraints, such as the space of support functions of convex bodies. As an example, we reconstruct below the support function of the icosaedron:
The proximal algorithm can also be applied to this modified case, thus providing the first method able to approximate the projection of a function on the sphere onto the space of support functions of convex bodies. We present below the numerical computations of projections (for ) of the support function of a unit regular simplex onto the set of support functions of convex bodies with constant width. We believe that these projection operators could be useful in the numerical study of a famous conjecture due to Bonnesen and Fenchel (1934) concerning the so-called Meissner's convex bodies.