For definitions and reference, we are working from Guenter Ziegler's Lectures on Polytopes.
We define the free sum as follows. Given two polyhedra $P, Q \in \mathbb{R}^d$, the free sum $P \bigoplus Q = \text{ conv }(P \cup Q)$.
Here are my thoughts so far:
One definition of polyhedra is that we have some finite set of points $V_1$ and some finite list of vectors $Y_1$ such that $P = \text{ conv }(V_1) + \text{ cone }(Y_1)$. Similarly, $Q = \text{ conv }(V_2) + \text{ cone }(Y_2)$.
So, $$\text{ conv }(P \cup Q) = \text{ conv }((\text{ conv }(V_1) + \text{ cone }(Y_1)) \cup (\text{ conv }(V_2) + \text{ cone }(Y_2)))$$
I'd like to show that this can somehow reduce to our previous definition, so $P \bigoplus Q = \text{ conv }(V) + \text{ cone }(Y)$ for some finite $V,Y$.
But maybe this is not a good direction to go in proving this.