Monotone convergence of Riemannian gradient descent for the center of mass

Let $M$ be a Riemannian manifold, let $\rho\leq r_{\operatorname{cx}}$, and let $\{x_i\}_{i=1}^N\subset B(o,\rho)\subset M$. Set $\bar{x}_2$ to be their $L^2$ center of mass, and define $H_{B(o,\rho)}=c_{\delta}(2\rho)$, where $c_{\kappa}$ is the comparison function defined in the paper. Assume $x^0\in B(o,\rho)$ and use the constant step-size $t_k=t$ with $t\in(0,1/H_{B(o,\rho)}]$. Monotone convergence conjecture. Every iterate remains in $B(o,\rho)$, $f_2(x^{k+1})\leq f_2(x^k)$ for $k\geq0$, with equality only when $x^k=\bar{x}_2$, and $x^k\to\bar{x}_2$ as $k\to\infty$. More generally, for $2\leq p<\infty$, the same conclusions hold when $t\in(0,1/H_{B(o,\rho),p}]$, where $H_{B(o,\rho),p}=(2\rho)^{p-2}\max\{p-1,c_{\delta}(2\rho)\}$. These conditions give a proposed sharp convergence regime for the algorithm, strengthening the paper's general sufficient convergence results.