Apologies if this is fundamental – I wonder where to find a proof and a reference for the following facts, which, I am sure, must be true.

(1) Suppose we are given a finite set of points in $ mathbb {R} ^ {d + 1} $. For each point, we are given a closed half-space that is tangent to it and strictly contains all the others. Then there is a strictly convex set contained in the intersection of half-spaces with all given points of the boundary.

(If I drop "strictly", then the convex hull of the dots works.)

(2) Suppose we are given a finite set of pairs $ (x, y) $ with $ x in mathbb {R} ^ d $ and $ y in mathbb {R} $. Suppose we are given a linear function in each point that is tangent and strictly inferior to the others. Then, there is a strictly convex function passing through the points whose gradient at each given point corresponds to the given linear function.

(If I drop "strictly", then the "convex closure" of the pairs works.)