Let $H$ be the Hermitians in $M_n$ and for norm one $B$ in $H$ define $T_B$ from $H$ to $M_n$ by $T_B(A)=AB$. Each $T_B$ is open onto its image and the assignment $B \mapsto T_B$ is continuous (by direct checking or because it is linear). From this it is easy to check that the degree of openness of $T_B$ is bounded away from zero for $B$ a norm one Hermitian.

From this it follows easily that such a $c_n$ exists.

(For the purpose of this post: if $T(Ball X) $ contains $a Ball (TX)$ say that the degree of openness of $T$ is at least $a$.)

Getting a good estimate for $c_n$ looks like a nice problem.

EDIT: As Denis so kindly pointed out, this answer is utter nonsense.

Let $H$ be the Hermitians in $M_n$ and for norm one $B$ in $H$ define $T_B$ from $H$ to $M_n$ by $T_B(A)=AB$. Each $T_B$ is open onto its image and the assignment $B \mapsto T_B$ is continuous (by direct checking or because it is linear). From this it is easy to check that the degree of openness of $T_B$ is bounded away from zero for $B$ a norm one Hermitian.

From this it follows easily that such a $c_n$ exists.

(For the purpose of this post: if $T(Ball X) $ contains $a Ball (TX)$ say that the degree of openness of $T$ is at least $a$.)

Getting a good estimate for $c_n$ looks like a nice problem.

EDIT: As Denis so kindly pointed out, this answer is utter nonsense.

Let $H$ be the Hermitians in $M_n$ and for norm one $B$ in $H$ define $T_B$ from $H$ to $M_n$ by $T_B(A)=AB$. Each $T_B$ is open onto its image and the assignment $B \mapsto T_B$ is continuous (by direct checking or because it is linear). From this it is easy to check that the degree of openness of $T_B$ is bounded away from zero for $B$ a norm one Hermitian.

From this it follows easily that such a $c_n$ exists.

(For the purpose of this post: if $T(Ball X) $ contains $a Ball (TX)$ say that the degree of openness of $T$ is at least $a$.)

Getting a good estimate for $c_n$ looks like a nice problem.

Let $H$ be the Hermitians in $M_n$ and for norm one $B$ in $H$ define $T_B$ from $H$ to $M_n$ by $T_B(A)=AB$. Each $T_B$ is open onto its image and the assignment $B \mapsto T_B$ is continuous (by direct checking or because it is linear). From this it is easy to check that the degree of openness of $T_B$ is bounded away from zero for $B$ a norm one Hermitian.

From this it follows easily that such a $c_n$ exists.

(For the purpose of this post: if $T(Ball X) $ contains $a Ball (TX)$ say that the degree of openness of $T$ is at least $a$.)

Getting a good estimate for $c_n$ looks like a nice problem.

EDIT: As Denis so kindly pointed out, this answer is utter nonsense.