# Comparing volumes of $d$-dimensional unit-balls to upper bound kissing-number.

I want to analyze so called kissing-numbers in $$d$$-dimensional euclidean-spaces.

The kissing number $$\kappa(d)$$ of a dimension $$d$$ is defined as the greatest number of non-overlapping unit $$d$$-spheres that can be arranged in the space such that they each touch a common unit $$d$$-sphere.

Therefore consider a $$d$$-dimensional unit-ball in the euclidean space. It is obvious that it can only be touched by a finite number of other unit-balls. Also, since we use unit balls, all balls of a maximal kissing-configuration are contained in another ball of diameter $$3$$.

Can you show that the total number of balls in such a configuration, namely $$\kappa(d)+1$$ balls, can be upper bounded by $$3^d$$.

So my goal is to show: $$\kappa(d)+1\leq 3^d$$ where $$d$$ is the dimension.

For example could one show this by comparing volumes? If so, how?

This can indeed be done by comparing volumes. The volume $$V_d(r)$$ of a $$d$$-dimensional sphere of radius $$r$$ is $$V_d(r)=C_dr^d,$$ for some positive constant $$C_d$$ independent of $$r$$. It follows that the number of pairwise disjoint unit $$d$$-spheres in a $$d$$-sphere of radius $$3$$ is at most $$\frac{V_d(3)}{V_d(1)}=\frac{C_d3^d}{C_d1^d}=3^d,$$ and hence the kissing number $$\kappa(d)$$ in dimension $$d$$ satisfies $$\kappa(d)+1\leq 3^d$$, as desired.