One campus, seven cafeterias

Here is a puzzle a friend asked me yesterday evening at Stanford. Might have been Palo Alto actually. Anyway, here it goes. It is in my own words. The mathematical problem is the same although the wording has been changed.

I solved it. Now, you try!

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College ThingamajigANameIForget on Mars is building seven new cafeterias. The college campus is a perfect planar circle with a radius of 1 km. The college wants to build the cafeteria so that each of them is located within the planar circle. The cafeterias should be so arranged as to minimize the maximum distance any student would have to travel to reach a cafeteria given that the student is located on campus.

In mathematical terms: you are given a circle C of radius 1 km and seven points. Arrange the seven points inside the circle so that you solve the following constrained optimization problem.

Let d(P, i) denote the linear distance between point P and cafeteria i.
Let dist(P) = min(d(P,i) for i ranging from 1 to 6)

That is, dist(P) is the minimum distance a student located at P would have to travel to reach some cafeteria.

Arrange the 7 cafeterias such that dist(P) is minimized when P is ranged over all points within the circle C.

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Update: fixed typo in post.