Finite Convex Integration

This paper is concerned with the finite convex integration problem which for a given finite family of pairs of points

, consists in finding a convex function f such that

x_i^*\in\partial f(x_i) x i ∗ ∈ ∂ f ( x i )

for all i, where

\partial f(x_i) ∂ f ( x i )

denotes the Fenchel-subdifferential of f at x