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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?

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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.

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