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

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 $\rho_1,\rho_2,...,\rho_m$, we show 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 combinatorial complexity $\Omega(n_1n_2^{\lfloor\frac{d}{2}\rfloor}+n_2n_1^{\lfloor\frac{d}{2}\rfloor})$. 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 problem of computing the worst-case combinatorial complexity of the convex hull of a set of $m$ $d$-dimensional convex polytopes lying on $m$ parallel hyperplanes in $\mathbb{E}^{d+1}$, where $d\ge{}3$ odd, a problem which is of independent interest. More precisely, we show that the worst-case combinatorial complexity of 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 for the parallel polytope convex hull problem, yield tight bounds on the combinatorial complexity of the Minkowski sum of two convex polytopes in $\mathbb{E}^d$.Comment: 22 pages, 5 figures, new proof of upper bound for the complexity of the convex hull of parallel polytopes (the new proof gives upper bounds for all face numbers of the convex hull of the parallel polytopes

Incremental and Decremental Maintenance of Planar Width

We present an algorithm for maintaining the width of a planar point set dynamically, as points are inserted or deleted. Our algorithm takes time O(kn^epsilon) per update, where k is the amount of change the update causes in the convex hull, n is the number of points in the set, and epsilon is any arbitrarily small constant. For incremental or decremental update sequences, the amortized time per update is O(n^epsilon).Comment: 7 pages; 2 figures. A preliminary version of this paper was presented at the 10th ACM/SIAM Symp. Discrete Algorithms (SODA '99); this is the journal version, and will appear in J. Algorithm