im new to the forum and I have a question that Im working quite for a while now.
I would like to prove that the greatest norm when considering the exponential $\exp{A(\alpha)}$, being $A(\alpha)$ the convex combination of $m$ real matrices, that is $A(\alpha) = \alpha_1 A_1+\alpha_2A_2+\dots+\alpha_mA_m$, $A_m \in \mathbb{R}^{n\times n}$, $\alpha_1 +\alpha_2+\dots+\alpha_m=1$, happens in one of the polytopic vertices, that is $ \max||\exp{A(\alpha)}|| = ||\exp{A_k}|| $, with $A_k$ one of the polytopic vertices.
I know for a fact that the solution of a convex maximization problem, given that the feasible region is polytopic, is always due to a vertex, but I can`t proceed to prove the convexity of $\exp{A(\alpha)}$. If I can proceed to prove the convexity of $\exp{A(\alpha)}$ than the composition with the norm function (which indeed is convex) will provide enough argument.
In my attempts to tackle the problem I have expanded the term $\exp{A(\alpha)}$ but couldn`t figure it out so far.