Let $\mathcal{R}$ be an $n$-dimensional bounded hyperrectangle, and consider a $n\times n$ matrix $A$ with real entries.
Given set $\mathcal{R}$ and matrix $A$, I want to compute or find an upper bound for the diameter of the set $\mathcal{X}$ defined by the set-valued integral \begin{eqnarray*} \mathcal{X} = \displaystyle\int_{a}^{b} \exp(tA)\:\mathcal{R}\:\mathrm{d}t, \qquad 0<a<b. \end{eqnarray*}
Trouble: $\mathcal{X}$ is a convex set but it is not clear to me whether it is a polytope.
To see this, recall the definition of point-to-set function: $$\int_{a}^{b}F(t)\mathrm{d}t = \displaystyle\lim_{\Delta\rightarrow 0} \displaystyle\sum_{i=\lfloor a/\Delta\rfloor}^{\lfloor b/\Delta\rfloor}\Delta F(i\Delta),$$ where the summation inside the limit is a Minkowski sum.
This leads us to $\displaystyle\sum_{i}\Delta \exp(i\Delta)\mathcal{R}$.
Each term in this sum is a (different) linear transformation of $\mathcal{R}$, hence the sum is a polytope but it is not clear to me if the limit can smooth the vertices out.
Even if the algebraic description of $\mathcal{X}$ is not possible, the diameter or a bound on its diameter will do for my purpose.