We prove the existence of optimal separators for intersection graphs of balls and spheres in any dimension $d$. One of our results is that if an intersection graph of $n$ spheres in $\mathbb{R}^d$ has $m$ edges, then it contains a balanced separator of size $O_d(m^{1/d}n^{1-2/d})$. This bound is best possible in terms of the parameters involved. The same result holds if the balls and spheres are replaced by fat convex bodies and their boundaries.

翻译：我们证明了在任意维度 $d$ 下，球体和球面交集图存在最优分隔符。其中一项结果是：若 $\mathbb{R}^d$ 中 $n$ 个球体的交集图有 $m$ 条边，则它包含一个大小为 $O_d(m^{1/d}n^{1-2/d})$ 的平衡分隔符。该界在涉及参数的意义上是最优的。当球体和球面被替换为胖凸体及其边界时，相同结论依然成立。