Is the exponential image of a convex set of matrices convex?

Question

Let $\Omega \subset \mathbb{C}^{n \times n}$ be a closed convex set of $n \times n$ matrices (can be restricted to Hermitian matrices).

Let $\Gamma = e^\Omega = \{ Z ; \;\exists X\in \Omega \text{ and } Z = e^X \}$ where $e^X$ is the matrix exponential.

Is $\Gamma$ a (closed) convex set as well? If yes, what is the argument?

Thanks

Answer 1

Let $\Omega$ be the set of diagonal matrices of the form $$ A = \pmatrix{\lambda &0 \\0&1-\lambda}, \ \lambda\in[0,1]. $$ Then $e^\Omega$ is the set of matrices of the form $$ \pmatrix{e^\lambda &0 \\0&e^{1-\lambda}}, \ \lambda\in[0,1], $$ which is not convex.

To prove that such a set is not necessarily closed take $n=1$, $\Omega=(-\infty,0]$, $e^\Omega = (0,1]$.