Allometric relationships between traveltime channel networks, convex hulls, and convexity measures

Allometric relationships between traveltime channel networks, convex hulls, and convexity measures

[1] The channel network (S) is a nonconvex set, while its basin [C(S)] is convex. We remove open-end points of the channel connectivity network iteratively to generate a traveltime sequence of networks (Sn). The convex hulls of these traveltime networks provide an interesting topological quantity, which has not been noted thus far. We compute lengths of shrinking traveltime networks L(Sn) and a...

Approximate Convexity and Submonotonicity in Locally Convex Spaces

Formalizing Convex Hulls Algorithms

We study the development of formally proved algorithms for computational geometry. The result of this work is a formal description of the basic principles that make convex hull algorithms work and two programs that implement convex hull computation and have been automatically obtained from formally verified mathematical proofs. A special attention has been given to handling degenerated cases th...

Convex hulls, oracles, and homology

This paper presents a new algorithm for the convex hull problem, which is based on a reduction to a combinatorial decision problem CompletenessC, which in turn can be solved by a simplicial homology computation. Like other convex hull algorithms, our algorithm is polynomial (in the size of input plus output) for simplicial or simple input. We show that the “no”-case of CompletenessC has a certi...

Steaming on Convex Hulls

This is a sea story about using a simple classroom example to save a great deal of money, as well as to convince beginning Postgraduate Naval School operations research students—experienced, skeptical military officers—that mathematical analysis can yield immediate results. The application is planning a ship’s transit from one point to another in a fixed amount of time, given that the ship can ...