Small-dimensional linear programming and convex hulls made easy

Given a set @S of spheres in E^d, with d>=3 and d odd, having a constant number of m distinct radii @r"1,@r"2,...,@r"m, we show that the worst-case combinatorial complexity of the convex hull of @S is @Q(@__ __"1"=<"i"<>"j"=<"mn"in"j^@__ __^d^2^@__ __), ...

Given a set Σ of spheres in E^d, with d≥3 and d odd, having a fixed number of m distinct radii ρ₁,ρ₂,...,ρ_m, we show that the worst-case combinatorial complexity of the convex hull CH_d(Σ) of Σ is Θ(Σ_{{1≤i≠j≤m}}n_in_j^{⌊ d/2 ⌋}), where n_i is the number of ...

We present two randomized algorithms. One solves linear programs involvingm constraints ind variables in expected timeO(m). The other constructs convex hulls ofn points in d,d>3, in expected timeO(n[d/2]). In both boundsd is considered to be a constant. ...