Let’s say a country has 5 major cities. What should the shape of the country look like such that each city is located **closest to each other** as possible ? Similarly what would the shapes be for the cases with 6, 7 , 8 cities?

A very small pentagon. Unless we can stack cities.

Should the territory of each state (or province,parish, county, whatever word you use ) should be as close as possible to its city too ? That would change things.

You’d a scoring system to balance the competing issues of distance between city CBD’s and the distance of each part of the cities larger area to its CBD…

But it all depends on the shape of the country too !

I don’t understand the question. On the assumption that cities can be located anywhere within the country, then no matter what shape (or size) the country is, five cities (or any number of cities) can all be adjacent to one another.

I’m afraid you’re leaving out part of the problem. Regardless of the country’s shape you can minimize the distances among 5 cities by making them arbitrarily close to each other, as with New York’s 5 largest:

♪♫ *Give me Manhattan, the Bronx and Staten Island too…* ♪ ♫

There are some famous related problems. Here’s one: Position 8 points on a sphere to maximize the shortest distance between any two. (The solution is *not* a cube arrangement.)

I cheated, but cool, not a cube … I hope you feel sorry for everyone I talk to today [grin].

Without cheating:

Start with points at the corners of a cube. Rotate the top square of points 45 degrees (in the plane of those four points) relative to the bottom four points. All the points on two different squares are now farther apart than two points sharing an edge in a single square, so you can move the two squares a little close together, and make them a little bigger. They’ll be farther apart than the cube arrangement, but I can’t prove that it’s maximal.

Well done! The shape is called a square antiprism.

Here are pictures of the square antiprism and related solutions with a link at the bottom to an archived version of one of N. J. A Sloane’s pages.

I suppose it’s possible optimality has never been proven – after all the “obvious” Kepler’s Conjecture was proven only recently.

You mean "♪♫ *We’ll have . . . "*

No, I think you mean "♪♫ *I’ll take . . . "*

:smack: That’s not the first time I’ve gotten song lyrics wrong; I [del]am afraid[/del] hope it won’t be the last.

My excuse is that I’m not quite old enough to have seen *The Garrick Gaieties*. I think I heard the song in an episode of *Kojak*. Maybe Telly Savalas missang the lyrics.

You wouldn’t need to stack the cities. If you calculate city distances using the center of mass of each city, you could gerrymander each city’s borders such that they’re all at the same spot in the country. The borders would need to interlace, but not necessarily overlap.

This achieves a more optimal solution than having more simply shaped cities adjacent to each other.

As previously mentioned, the shape of the country is most likely irrelevant.

The simplest way to do that would be to make the cities concentric rings, a sort of bulls-eye pattern.