Is the minimizer of condition number of a PSD matrix $A$ the same as the minimizer of $\lambda_{max}(A) - \lambda_{min}(A)$? Given a positive semidefinite matrix $A\in\mathcal{C}\subset\mathbb{S}^n_+$, where $\mathcal{C}$ is a convex set. Define the following minimizers:
$$
\bar{A} = \arg\min_{A\in\mathcal{C}} \frac{\lambda_{max}(A)}{\lambda_{min}(A)}, \quad \tilde{A} =\arg\min_{A\in\mathcal{C}} \lambda_{max}(A) - \lambda_{min}(A)
$$
$\bar{A}$ is the minimizer of the ratio of largest eigenvalue and smallest eigenvalue; while $\tilde{A}$ is the minimizer of the difference between the largest eigenvalue and smallest eigenvalue. Are the two minimizer equals, i.e., $\bar{A} = \tilde{A}$?
I presume they are not the same in general and perhaps depends on the convex set $\mathcal{C}$. But I think they are somehow connected, as these two objective functions are both trying to bring the largest and smallest eigenvalues closer together.

You can embed any polytope $P\subset\mathbb{R}^n_+$ as a convex set in $\mathbb{S}_+$ via the embedding $(x_1,x_2,\dots,x_n)\mapsto diag(x_1,x_2,\dots,x_n)$. Let $\mathcal{C}$ be the image of $P$ under this embedding. Then, the quantities in question in the polytope world become $$
\kappa_1(P) = \arg\min_{x\in P} \frac{\max_i x_i}{\min_i x_i},\quad\quad \kappa_2(P) = \arg\min_{x\in P}\;\big( (\max_i x_i)-(\min_i x_i)\big).
$$
These two minimisers are not necessarily equal. A very simple example is the following: $$
P = \{(x,y)\in\mathbb{R}^2_+\mid x+y=1 \}.
$$ Then $\kappa_1(P) = (0.5,0.5)$ but $\kappa_2(P)=(1,0)$ or $(0,1)$.
