# Kissing number lower bound vs. upper bound - precise meanings?

According to en.wikipedia.org, https://en.wikipedia.org/wiki/Kissing_number#Some_known_bounds

It says the kissing numbers $$K$$ have lower bound $$K_L$$ and upper bound $$K_S$$: $$K_L < K < K_U.$$

I am slightly puzzled by what these lower bounds and upper bounds really constrain.

Here are my puzzles:

1. Isn't that the lower bound can be just 0? The ball does not kiss with its nearest neighbors?

2. Suppose the goal is to achieve the densest sphere packing - then shouldn't the solution always be the highest kissing number $$K_U$$? What are the exceptions?

3. Are there examples of the lower bound kissing numbers $$K_L$$ that can give rise to the densest sphere packing?

4. Only when the solutions are found, then the $$K_L=K_U$$ always? What exactly is the logic here? Why not $$K_L \neq K_U$$ and $$K_L < K < K_U$$ at dimensions = 1,2,3,8,24?

5. How about dimension = 4? Why $$K_L = K = K_U = 24$$?

Thanks!

• Maybe you'd be interested in the paper "A survey on the kissing numbers" by Peter Boyvalenkov, Stefan Dodunekov, and Oleg R. Musin: arxiv.org/abs/1507.03631 May 29 at 20:38

Lots of questions here, I'll see how many I can address. To be clear, $$K_L$$ and $$K_U$$ are summaries of our current knowledge. There is some true kissing number in each dimension, and hopefully we'll learn what it is someday; this is just what we know now.
(1) The $$n$$-dimensional kissing number, $$K$$, is defined to be the maximum number of non-overlapping spheres capable of touching an $$S^{n-1}$$. So saying that $$K_L \leq K$$ means that we have found a way to place $$K_L$$ many non-overlapping spheres so that they touch an $$S^{n-1}$$. Of course, we could place fewer spheres in contact with a sphere (as you say, we could place $$0$$), but $$K_L$$ is the largest number of spheres we have managed to place.
(2) Probably not. The conjectured best packings in $$9$$ and $$10$$ dimensions are the $$9$$ and $$10$$ dimensional laminated lattices, with kissing numbers $$272$$ and $$336$$ (see here) and the conjectured best lattice in $$12$$ dimensions is the Coxeter-Todd lattice with kissing number $$756$$; these are less than the numbers in your table. I see no reason that the densest packing should also have the highest kissing number, but since we have very little understanding of either problem, I don't know if we can rule it out.
(3), (4) and (5): The dimensions $$1$$, $$2$$, $$8$$ and $$24$$ are very special, because we know exactly what the densest lattice is and it also achieves the maximum kissing number. In dimension $$3$$, the Kepler conjecture, now proved by Hales, establishes that the optimal density is achieved by the face centered cubic lattice, which indeed has kissing number $$12$$. I don't know what is going on in dimension $$4$$.
Again, what is going on here is that there is special structure which allows us to know $$K$$ exactly in these cases, and so $$K_L = K_U$$. The other cases where $$K_L < K_U$$ in the table, reflect our current ignorance, not eternal mathematical truth.