I am interested in convex functions on the space of convex polytopes.
Let $\mathbb{R}^n$ denote the $n$-dimensional Euclidean space. A convex polytope is the convex hull of a finite subset of $\mathbb{R}^n$. Let $\mathcal{P}$ denote the space of all convex polytopes. Endow $\mathcal{P}$ with the following linear combination operation:
$$\big(\forall A \in \mathcal{P} \big) \big(\forall B \in \mathcal{P} \big) \big(\forall \lambda \in [0,1] \big) \big( \lambda A + (1-\lambda) B \equiv \{\lambda x+(1-\lambda)y:x\in A,y\in B \} \big)$$
I am interested in convex and real-valued functions on $\mathcal{P}$. That is, $f:\mathcal{P} \to \mathbb{R}$ such that
$$f(\lambda A+(1-\lambda)B)\leq\lambda f(A)+(1-\lambda)f(B)\ \forall\ A,B\in\mathcal{P} \text{ and } \lambda\in(0,1)$$
I would like to know how to characterize such convex functions. Does anyone know any reference?
I have found several characterizations for convex functions defined on Euclidean space. But here the domain is very different. The space of convex polytopes $\mathcal{P}$ is not even a vector space.
This should allow us to prove that $f$ is convex everywhere, since for any length $l$, I can create a box $l$ polytope, and the function $f$ will be convex on this box.
Since this works for arbitrarily large boxes, it will work over $\mathbb{R}^n$. Perform a proof by contradiction: Assume the function is not convex in some interval, then take a box that covers that interval.
So, this is equivalent to "convex function over $\mathbb{R}^n$ as far as I can tell :)