I wonder if something similar to the following fact is known, and I would greatly appreciate any references.
Let $t_1, t_2, \ldots, t_N$ be unit vectors in $\mathbb{R}^n$.
Let $S$ denote the unit sphere in $\mathbb{R}^n$. For each $t_i$, define the maximal inner product region set: $$ A_i = \left\{ x \in S \mid \langle x, t_i \rangle > \langle x, t_k \rangle \text{ for all } k = 1, \dots, N \right\}. $$ The union of $A_i$ covers $S$, except for a set of measure zero.
Define $B_i$ as the reflection of $A_i$ with respect to the hyperplane orthogonal to $t_i$: $$ B_i = \left\{ x - 2 \langle x, t_i \rangle t_i \mid x \in A_i \right\}. $$ In general, the union of $B_i$ does not cover $S$ in the sense that $A_i$ do.
However, it seems the fraction of the measure occupied by $\bigcup_i B_i$ is not too small. I suspect it is a constant fraction, but references to even weaker results (say, $O(n^{-k})$) are welcome. Can we bound the measure of $\bigcup_i B_i$ from below?
It could be helpful (and could lead to a stronger estimate) to put assumptions on the collections $t_1,\ldots,t_n$. For instance, one could assume that $t_i$ form an $\varepsilon$-net on $S$ and $N$ is large.
