Let $P$ be a convex polytope (or more generally convex body, I suppose) in $V=\mathbb{R}^n$. For each $v\in \mathbb{P}V$, we define an involution $\tau_v\colon P\to P$ by setting $\tau_v(p)$ to be the reflection of $p$ within the line segment in $P$ passing through $p$ in direction $v$.

Let $p$ be a point in the interior of $P$. For each $k \geq 0$, let $\mu_k$ denote the probability distribution of $\tau_{v_k} \tau_{v_{k-1}} \cdots \tau_{v_1} p$ on $P$, where $v_1,\ldots,v_k \in \mathbb{P}V$ are chosen independently according to the standard distribution on real projective space. Does this sequence of distributions $\mu_0,\mu_1,\ldots$ converge to some measure in the limit (I assume so…)? What measure does it converge to (it is not uniform, is it...)? In general, is there somewhere these kind of random walks are studied?

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