Midpoints on Riemannian manifold with bounded curvature Let $M$ be a Riemannian manifold with non-negative sectional curvature everywhere bounded above by some $\kappa>0$. For simplicity, assume that the injectivity radius is everywhere infinite on $M$, so the log map at any $x\in M$, denoted by $Log_x$, is defined everywhere on $M$.
Now, let $x_1,x_2$ be fixed points in $M$.
For any $x\in M$, I want to quantify the difference between :

*

*taking the midpoint of $Log_x(x_1)$ and $Log_x(x_2)$ in the tangent space $T_xM$ and map it back to $M$, and

*taking the midpoint $x^*$ of $x_1$ and $x_2$, defined as the minimizer of $d(y,x_1)^2+d(y,x_2)^2$ over $y\in M$, where $d$ is the geodesic distance.

In other words, can we bound
$$\left\|Log_x(x^*)-\frac{1}{2}\left(Log_x(x_1)+Log_x(x_2)\right)\right\|_x^2$$
by a quantity that only depends on $\kappa$ and $d(x,x^*)$ ?
And what about the case where $M$ has non-positive sectional curvature everywhere bounded below by $-\kappa$ (for some $\kappa>0$) ?
 A: For fixed $x_1,x_2$ one does get a bound for above depending only on $\kappa$ and $d(x,x^*)$. Let me only consider $x$ for which $d(x,x^*)<\frac{1}{4}d(x_1,x_2)$. For other choices of $x$, triangle inequality provides the upper bound $d(x,x^*)$ upto a constant. Now for $d(x,x^*)<\frac{1}{4}d(x_1,x_2)$, first one observes $$\frac{1}{4}d(x_1,x_2)\leq d(x,x_1)\leq \frac{3}{4}d(x_1,x_2),$$ again by triangle inequality and similarly for $d(x,x_2)$. Let $\theta_x$ be the angle between $\log_x(x_1)$ and $\log_x(x_2)$. Then $$\|\frac{1}{2}(\log_x(x_1)+\log_x(x_2))\|=\frac{1}{2}\sqrt{d(x,x_1)^2+d(x,x_2)^2+2d(x,x_1)\cdot d(x,x_2)\cos\theta_x}$$ and $\|\log_x(x^*)\|=d(x,x^*)$. One can use Toponogov's triangle comparison theorem to bound $\theta_x$ in terms of $\kappa$ and $d(x,x^*)$. Note that as $d(x,x^*)$ goes to zero, $\theta_x$ goes to $\pi$. The curvature lower bound says that the $\theta_x$ can not be too small.
Edit: In the case of strictly positive sectional curvature, note that Topogonov's theorem says that the angle between $\log_x(x_1)$ and $\log_x(x_2)$ can not be less than $\pi-\epsilon$ for any $\epsilon>0$ as $x\to x^*$ (you need one of the sides to be small, let it be the one joining $x$ and $x^*$). If the angle is greater than $\pi+\epsilon$ for some $\epsilon>0$, as $x\to x^*$, then again Arzela-Ascoli gives a contradiction to an infinite injectivity radius.
