Let $M$ be Riemannian manifold and $p_{1},p_{2}$ two points that we can suppose close enough to use normal coordinates.
My goal would be estimate the $1$-moment of the ball $B_{t}$ of center $p_{2}$ and radius $t$ at $p_{1}$, which is $$\int_{B_{t}}d(p_{1},x)d\mu$$
More precisely, ideally, I would like to get an aproximation of this moment depending on $d(p_{1},p_{2})$, the metric tensor at $p_{2}$ and some o$(t^{k})$.
My goal is to be able to compare this moment to $d(p_{1},p_{2})\times Vol(B_{t})$ when $d(p_{1},p_{2})$ is small and $t$ is small compared to $d(p_{1},p_{2})$.
EDIT : I will share some incomplete reasonning that might be improved to get to a solution.
We know that we can define the sectional curvature at a point $x$ and two unit tangent vectors $v,w$ as the quantity $K(v,w)$ such that we have:
$$d(\exp_{x}\epsilon w,\exp_{\exp_{x}\delta v}(\epsilon w+\delta v))=\delta(1-\epsilon^{2}\frac{1}{2}K(v,w))+O(\delta\epsilon^{3}+\delta^{2}\epsilon^{2})$$ where $\epsilon,\delta$ are small reals and $\epsilon w+\delta v$ denote the parrallel transport of $\epsilon w$ along $\delta v$.
$\exp_{x}\delta v$ is denoted Let us work under the (optimistic and unclear) assumption that there exists a constant $A$ such that the sectional curvature also check: $$d(\exp_{x}\epsilon w,\exp_{x}\delta v)=|\delta v-\epsilon w|(1-\epsilon^{2}AK(v,w))+O(\text{smaller expression of $\epsilon,\delta\to 0$})$$
Coming back to our case, here $x$ would be the center $p$ of the small ball $B(p,r)$, $\delta v$ would be the antecedant of $q$ by the exponentiel at $p$ and $\exp_{x}\epsilon w=\exp_{p}\epsilon w$ would vary inside the ball $B(p,r)$
Therefore, the moment at $q$ end up as: $$\int_{u\in B_{r}}|\delta v-u|(1-\epsilon^{2}AK(v,\frac{u}{|u|}))+(\text{small terms in $\delta,r\to 0$})$$
were $B_{r}$ denote the euclidian ball of radius $r$ at the origin.
Denote $$E=\int_{u\in B_{r}}|\delta v-u|$$ $E$ is the euclidian case of the problem.
Then the 1-moment $M$ of $B(p,r)$ at $q$ become: $$M=E-\int_{u\in B_{r}}|\delta v-u|\epsilon^{2}AK(v,\frac{u}{|u|})+(\text{small terms in $\delta,r\to 0$})$$
Now if we denote $R=\int_{u\in B_{r}}|\delta v-u|\epsilon^{2}AK(v,\frac{u}{|u|})$ so that: $$M=E-R+\text{(smaller terms in $\delta,r\to 0$)}$$ under which hypothesis can I obtain a control of $R$ in $\delta,r$ relatively to $E$, for example, is there hypothesis on $Ricc_{x}(v)$ that allow to put $R$ in the "smaller term in $\delta,r\to 0$" ?
And also, am I too bold to assume the existence of the constant $A$ ?