**Christmas tree problem**You have a fir tree and want to decorate it with christmas lights.

The fir tree's green bushy "surface" is cone-shaped, with a circular boundary at the bottom.

You also have

*n* candles. These are about to be placed on the fir tree.

In reality, the candles would be connected by electrical cable, which would restrict the candles' distribution on the tree. Assume that the candles need no electricity, or that the cable between any two candles is arbitrarily long.

**Task.** Distribute the

*n* candles on the fir tree such that the candles are nicely spaced apart from all other candles. You may place candles on the boundary.

Precisely: Given a distribution

*D* of the candles, compute the minimum of all distances

*d*(

*k*,

*k*') between all candles

*k* ≠

*k'* of

*D*. We call this minimum

*f*(

*D*). We want to maximize

*f*. Thus, denote by

*M* the supremum of the minimum distances of all distributions,

*M* = sup {

*f*(

*D*) :

*D* is a distribution }. Finally, find an arrangement of the candles

*D* that attains

*f*(

*D*) =

*M*. If no such arrangement exist, instead find a sequence (

*D*_{i}) of arrangements such that the sequence of minimums

*f*(

*D*_{i}) converges to

*M*.

In a variations of the christmas tree problem, you're encouraged to avoid the boundary, too. Then, for each distribution, don't merely compute the minimum of all candle-to-candle distances, but also the minimum of all candle-to-boundary distances, and use the minimum of these two minimums. Across all distributions, maximize this value.

**Example.** Given only one candle, put it anywhere you like. If you'd like to avoid the boundary, too, then put the lone candle at the top of the tree.

**Example.** Given two candles, even if distance to the boundary doesn't matter, the best distribution depends on the shape of the cone. If the cone is very tall and thin, put one candle at the top and one on the circular boundary at the bottom. Otherwise, if the cone is squat and low-risen, put both candles on opposite points of the boundary.

**Example.** If your christmas tree is not a tree at all, but rather the unbounded real line, you can space the candles arbitrarily far from each other. This is very nice.

**Application.** If your christmas tree is instead the space of all colors, and you have 8 Lix player colors, find a distribution of 8 colors such that no two colors look more similar than necessary. This is hard, especially if you, in addition, want to avoid black because black lixes looks too much like the

~~boundary~~ level background.

**End.** The christmas tree problem is really about abstract metric spaces, not christmas trees, but christmas trees inspired me to first think about this problem in detail. I don't know whether this problem is known by other names in mathematics. I don't know whether it's more common to avoid the boundary or not.

-- Simon