Let $G$ be a connected reductive group over a complete discrete valuation field with perfect residue field (or just a non-arch local field). Let $\mathcal{B}$ be its reduced Bruhat-Tits building, and $d(\cdot,\cdot)$ the distance function on $\mathcal{B}$. I am wondering if the following statement is true:
There exists a constant $C$ such that for any compact element $x\in G$ (i.e. $\{x^n|n\in\mathbb{Z}\}$ is contained in a compact subgroup), and any $p_1, p_2\in\mathcal{B}$ such that $x.p_1=p_2$, there exists $p_0\in\mathcal{B}$ with $$x.p_0=p_0,\;d(p_0,p_1)\le C\cdot d(p_1,p_2).$$
What I actually want is have an explicit $C$ in terms of $G$. For example, I'll guess that $C$ can be bounded by a constant multiple of the rank of $G$. E.g. when the rank is $1$ the assertion is true for $C=\frac{1}{2}$.
The motivation is for the following question, which in terms is for the interest of orbital integrals: Can we cover the ``compact'' part of a $G(\mathcal{O})$-double cosets by a bounded estimable number of conjugates of $G(\mathcal{O})$? Thanks!