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?