# Quasi concavity of max-relative entropy?

The max-relative entropy between two states is defined as

$$D_{\max }(\rho \| \sigma):=\log \min \{\lambda: \rho \leq \lambda \sigma\}.$$

It is known that the max-relative entropy is quasi-convex. That is, for $$\rho=\sum_{i \in I} p_{i} \rho_{i}$$ and $$\sigma=\sum_{i \in I} p_{i} \sigma_{i}$$ where $$\rho_i, \sigma_i$$ are quantum states and $$p$$ is a probability vector, it holds that

$$D_{\max }(\rho \| \sigma) \leq \max _{i \in I} D_{\max }\left(\rho_{i} \| \sigma_{i}\right).$$

Is there a lower bound for $$D_{\max }(\rho \| \sigma)$$ in terms of $$D_{\max }(\rho_i \| \sigma_i)$$?

No, this is not possible. Consider $$\rho_1 = \sigma_2 = \vert 0\rangle\langle 0 \vert$$ and $$\rho_2 = \sigma_1 = \vert 1\rangle\langle 1 \vert$$. Then,
$$D_{\max}(\rho_i\|\sigma_i) = \infty\quad \text{for } i = 1,2.$$
Let $$p_i = (1/2, 1/2)$$ and you see that $$D_{\max}(\rho\|\sigma) = 0$$.