  We introduce a sharpened version of the well-known Lonely Runner Conjecture
of Wills and Cusick. Given a real number $x$, let $\Vert x \Vert$ denote the
distance from $x$ to the nearest integer. For each set of positive integer
speeds $v_1, \ldots, v_n$, we define the associated maximum loneliness to be
$\text{ML}(v_1, \ldots, v_n)=\max_{t \in \mathbb{R}}\min_{1 \leq i \leq n}
\Vert tv_i \Vert$. The Lonely Runner Conjecture asserts that $\text{ML}(v_1,
\ldots, v_n) \geq 1/(n+1)$ for all choices of $v_1, \ldots, v_n$. If the Lonely
Runner Conjecture is true, then the quantity $1/(n+1)$ is the best possible,
for there are known equality cases with $\text{ML}(v_1, \ldots, v_n)=1/(n+1)$.
A natural but (to our knowledge) hitherto unasked question is: If $v_1, \ldots,
v_n$ satisfy the Lonely Runner Conjecture but are not an equality case, must
$\text{ML}(v_1, \ldots, v_n)$ be uniformly bounded away from $1/(n+1)$? We
conjecture that, contrary to what one might expect, this question has an
affirmative answer that reflects an underlying rigidity of the problem. More
precisely, we conjecture that for each choice of $v_1, \ldots, v_n$, we have
either $\text{ML}(v_1, \ldots, v_n)=s/(ns+1)$ for some $s \in \mathbb{N}$ or
$\text{ML}(v_1, \ldots, v_n) \geq 1/n$. Our main results are: confirming this
stronger conjecture for $n \leq 3$; and confirming it for $n=4$ and $n=6$ in
the case where one speed is much faster than the rest. We also obtain a number
of related results.
