Let A be a compact convex set in n-dimensional space. [ Of principal interest is n > 2 . ]
A result of Caratheodory states that A is equal to the union of its simplices (i.e. simplices with all (n+1) vertices lying in A ).
Suppose further that each of these simplices has volume bounded above by one .
Does this yield an upper bound for the volume of A itself?
Here's a crude but easy bound. Let $\triangle$ be a regular simplex of volume one centred at the origin; WLOG, $\triangle$ is the largest volume simplex contained in $A$. Then $A\subseteq -n\triangle$, so $\operatorname{vol}(A)\le n^n$.
(If $A$ is the Euclidean ball circumscribed around $\triangle$, it satisfies your condition, and $\operatorname{vol}(A)$ is like $(cn)^{n/2}$.)