From the paper "The operational meaning of min- and max-entropy", I am having trouble understanding the following sentence starting from "Since the optimization..."
Letting $\text{Pos}(\mathcal{H}_{X})$ denote the set of positive semi-definite matrices acting on $\mathcal{H}_{X}$, what I don't quite see is the compactness of the set $\{\sigma_B\in\text{Pos}(\mathcal{H}_{B}) \mid \text{id}_A\otimes\sigma_B - \rho_{AB}\in\text{Pos}(\mathcal{H}_{AB})\}$ over which the infimum on the left hand side is taken. The full article can be found here: https://arxiv.org/pdf/0807.1338.pdf
Any help is appreciated.
Its true that this particular set is not compact. However, where it is unbounded is in the direction away from the minimum.
So let $U = \{\sigma_B\in\text{Pos}(\mathcal H_B) \mid \text{id}_A\otimes\sigma_B - \rho_{AB}\in\text{Pos}(\mathcal H_{AB})\}$ and pick some $\sigma_0 \in U$. Then define $$U_0 := \{\sigma \in U\mid \operatorname{tr}(\sigma) \le \operatorname{tr}(\sigma_0)\} = U \cap \{ \sigma \in \mathcal H_B \mid \sigma \ge 0 \text{ and } \operatorname{tr}(\sigma) \le \operatorname{tr}(\sigma_0)\}$$
The latter set is compact and $U$ is closed, so $U_0$ is compact. And it is not hard to see that $$\inf\ \operatorname{tr}(U) = \max\ \operatorname{tr}(U_0) \in U_0 \subset U$$