I encountered the following claim regarding the strong convexity of Bregman divergence. Let $d$ be a sufficiently large integer. Let $p=1+1/\ln d\in (1,2]$ for sufficiently large $d$. For any $x,y\in\mathbb R^d_+$, define $D(x,y)$ as $$ D(x,y) := \frac{1}{p}\sum_{i=1}^d x_i^p - \sum_{i=1}^d x_i y_i^{p-1} + \left(1-\frac{1}{p}\right)\sum_{i=1}^d y_i^p. $$
Claim. There exists an absolute constant $0<c<\infty$ independent of $d$, such that for any $x,y\in\mathbb R_+^d$, $\|x\|_1,\|y\|_1\leq 1$, it holds that $$ D(x,y) \geq \frac{c}{\sqrt{\ln d}}\left[\sum_{i=1}^d |x_i-y_i|\right]^2. $$