Let $r>0$ and $X\in\mathbb{R}^{n\times n}$. Let $\|A\|$ denote any matrix norm of $A$. Is the ball $\mathcal{B}_r(X)\triangleq\{Y\in\mathbb{R}^{n\times n}:\|Y-X\|<r\} $ a convex set? How can I prove or disprove that?
I know that $\mathcal{B}_r(X)$ is convex if and only if for all $A,B\in\mathcal{B}_r(X)$, and all $\alpha\in[0,1]$, $$\|\alpha A+(1-\alpha)B-X\|< r.$$
Without a loss of generality, center the ball around zero. Then take $x,y\in \mathbb{R}^{n^2}$ with $||x||,||y||< r$. Then for $\lambda\in[0,1]$ we have $$ ||\lambda x+(1-\lambda)y||\\ \stackrel{\text{triangle ineq.}}{\leq} ||\lambda x||+||(1-\lambda)y||\\ \stackrel{\text{homogeneity}}{=}\lambda|| x||+(1-\lambda)||y||\\ <\lambda r+(1-\lambda)r=r $$