Positioning ice-cream stands on a street 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:

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*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.
 A: Pick the largest $t$ such that $2t<n$, and concentrate $f$ on $t$ small regions that are far apart.
Note that $t\ge \frac {n-2}2$.
If at least 3 stands get some share from a region, then at least one of them must be inside the region itself.
This implies that in this case only those stands can get any share that are very close to the region.
If a region has no stand very close to it, then assign the two closest stands to the region.
These are the only two stands that can get a share of that region.
By the pigeonhole principle, either there is a stand that is not assigned to, nor very close to any region (getting 0 share), or there is a region with 3 stands very close to it, none of which are assigned to any other region.
The share of one of these 3 stands will be at most $\frac1{3t}$, giving $g(n)\le \frac1{3t}\le \frac 2{3(n-2)}\approx \frac {1.5}n$.
This gives for example for $n=7$, by picking $t=3$, that $g(7)\le \frac 19$.
I believe that by playing a little with the distances between the regions one can get better bounds.
