On the number of tetrahedra with minimum, unit, and distinct volumes in three-space

We formulate and give partial answers to several combinatorial problems on volumes of simplices determined by $n$ points in 3-space, and in general in $d$ dimensions. (i) The number of tetrahedra of minimum (nonzero) volume spanned by $n$ points in \RR^3 is at most ${2/3}n^3-O(n^2)$, and there are point sets for which this number is ${3/16}n^3-O(n^2)$. We also present an $O(n^3)$ time algorithm for reporting all tetrahedra of minimum nonzero volume, and thereby extend an algorithm of Edelsbrunner, O'Rourke, and Seidel. In general, for every k,d\in \NN, $1\leq k \leq d$, the maximum number of $k$-dimensional simplices of minimum (nonzero) volume spanned by $n$ points in \RR^d is $\Theta(n^k)$. (ii) The number of unit-volume tetrahedra determined by $n$ points in \RR^3 is $O(n^{7/2})$, and there are point sets for which this number is $\Omega(n^3 \log \log{n})$. (iii) For every d\in \NN, the minimum number of distinct volumes of all full-dimensional simplices determined by $n$ points in \RR^d, not all on a hyperplane, is $\Theta(n)$.Comment: 19 pages, 3 figures, a preliminary version has appeard in proceedings of the ACM-SIAM Symposium on Discrete Algorithms, 200

Constant-Factor Approximation for TSP with Disks

We revisit the traveling salesman problem with neighborhoods (TSPN) and present the first constant-ratio approximation for disks in the plane: Given a set of $n$ disks in the plane, a TSP tour whose length is at most $O(1)$ times the optimal can be computed in time that is polynomial in $n$. Our result is the first constant-ratio approximation for a class of planar convex bodies of arbitrary size and arbitrary intersections. In order to achieve a $O(1)$-approximation, we reduce the traveling salesman problem with disks, up to constant factors, to a minimum weight hitting set problem in a geometric hypergraph. The connection between TSPN and hitting sets in geometric hypergraphs, established here, is likely to have future applications.Comment: 14 pages, 3 figure

New bounds on the average distance from the Fermat-Weber center of a planar convex body

The Fermat-Weber center of a planar body $Q$ is a point in the plane from which the average distance to the points in $Q$ is minimal. We first show that for any convex body $Q$ in the plane, the average distance from the Fermat-Weber center of $Q$ to the points of $Q$ is larger than ${1/6} \cdot \Delta(Q)$, where $\Delta(Q)$ is the diameter of $Q$. This proves a conjecture of Carmi, Har-Peled and Katz. From the other direction, we prove that the same average distance is at most $\frac{2(4-\sqrt3)}{13} \cdot \Delta(Q) < 0.3490 \cdot \Delta(Q)$. The new bound substantially improves the previous bound of $\frac{2}{3 \sqrt3} \cdot \Delta(Q) \approx 0.3849 \cdot \Delta(Q)$ due to Abu-Affash and Katz, and brings us closer to the conjectured value of ${1/3} \cdot \Delta(Q)$. We also confirm the upper bound conjecture for centrally symmetric planar convex bodies.Comment: 13 pages, 2 figures. An earlier version (now obsolete): A. Dumitrescu and Cs. D. T\'oth: New bounds on the average distance from the Fermat-Weber center of a planar convex body, in Proceedings of the 20th International Symposium on Algorithms and Computation (ISAAC 2009), 2009, LNCS 5878, Springer, pp. 132-14