We want to position $n$ ice-cream stands on a street. Assume that the population on the street is modeled by a nonnegative integrable function $f$, and everyone goes to the nearest ice-cream stand. What is the largest fraction $g(n)$ such that for any $f$, we can place the stands so that every stand serves at least a $g(n)$ fraction of the population? (Assume that no three stands can be in the same position. If two stands are in the same position, they can agree that one takes customers from the left and the other from the right.)
This setting is similar to the Hoteling-Downs model for spatial competition. Here are some observations:
It is easy to see that $g(1)=1$ and $g(2)=1/2$. By some casework, one can show that $g(3)=1/4$ and $g(4)=1/5$.
$g(n)\ge\frac{1}{2n}$ for all $n$. This is because we can divide the street into $n$ segments with equal population. For each segment, look at which half contains more population, and place a stand at the end of that half. (If both halves contain equal population, just choose any half.)
$g(n)\le\frac{1}{n+1}$ for all $n$. For odd $n$, consider when half of the population is concentrated at one end of the street and the other half at the other end. For even $n=2k$, consider when $\frac{k}{2k+1}$ of the population is at each of the two ends, and the remaining $\frac{1}{2k+1}$ just next to the midpoint.
