Is the Kullback-Leibler divergence *strictly* convex in both $p$ and $q$?

Question

The Kullback-Leibler divergence is convex in both $p,q$, where

$$\mathrm{KL}(p||q) = \sum_i p_i \ln \frac{p_i}{q_i}$$

for discrete probability distributions $p_i,q_i$.

Is it strictly convex?

Answer 1

No, it is not:

Let $p$, $q$ be two distinct discrete probability distributions on the same alphabet. Then,

$\operatorname{KL}(\lambda p + (1-\lambda) q \Vert \lambda p + (1-\lambda) q) = \lambda \operatorname{KL}(p \Vert p) + (1-\lambda) \operatorname{KL}(q \Vert q) = 0$

for any $\lambda \in [0,1]$.

However, $\operatorname{KL}(p \Vert q)$ is strictly convex in $p$ for fixed $q$, where it is finite. It is also strictly convex in $q$ for fixed $p$ if $\operatorname{supp}(q) \subseteq \operatorname{supp}(p)$. The proof follows from the condition for equality in the log-sum inequality.