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Given a set $\Sigma$ of spheres in $\mathbb{E}^d$, with $d\ge{}3$ and $d$odd, having a fixed number of $m$ distinct radii $\rho_1,\rho_2,...,\rho_m$, weshow that the worst-case combinatorial complexity of the convex hull$CH_d(\Sigma)$ of $\Sigma$ is$\Theta(\sum_{1\le{}i\ne{}j\le{}m}n_in_j^{\lfloor\frac{d}{2}\rfloor})$, where$n_i$ is the number of spheres in $\Sigma$ with radius $\rho_i$.  To prove the lower bound, we construct a set of $\Theta(n_1+n_2)$ spheres in$\mathbb{E}^d$, with $d\ge{}3$ odd, where $n_i$ spheres have radius $\rho_i$,$i=1,2$, and $\rho_2\ne\rho_1$, such that their convex hull has combinatorialcomplexity$\Omega(n_1n_2^{\lfloor\frac{d}{2}\rfloor}+n_2n_1^{\lfloor\frac{d}{2}\rfloor})$.Our construction is then generalized to the case where the spheres have$m\ge{}3$ distinct radii.  For the upper bound, we reduce the sphere convex hull problem to the problemof computing the worst-case combinatorial complexity of the convex hull of aset of $m$ $d$-dimensional convex polytopes lying on $m$ parallel hyperplanesin $\mathbb{E}^{d+1}$, where $d\ge{}3$ odd, a problem which is of independentinterest. More precisely, we show that the worst-case combinatorial complexityof the convex hull of a set $\{\mathcal{P}_1,\mathcal{P}_2,...,\mathcal{P}_m\}$of $m$ $d$-dimensional convex polytopes lying on $m$ parallel hyperplanes of$\mathbb{E}^{d+1}$ is$O(\sum_{1\le{}i\ne{}j\le{}m}n_in_j^{\lfloor\frac{d}{2}\rfloor})$, where $n_i$is the number of vertices of $\mathcal{P}_i$.  We end with algorithmic considerations, and we show how our tight bounds forthe parallel polytope convex hull problem, yield tight bounds on thecombinatorial complexity of the Minkowski sum of two convex polytopes in$\mathbb{E}^d$.

Convex hulls of spheres and convex hulls of convex polytopes lying on parallel hyperplanes