Abstract. The goal of this paper is twofold; first, show the equivalence between certain problems in geometry, such as view-obstruction and billiard ball motions, with the estimation of covering radii of lattice zonotopes. Second, we will estimate upper bounds of said radii by virtue of the Flatness Theorem. These problems are similar in nature with the famous lonely runner conjecture.
Minkowski's 2nd theorem in the Geometry of Numbers provides optimal upper and lower bounds for the volume of a o-symmetric convex body in terms of its successive minima. In this paper we study extensions of this theorem from two different points of view: either relaxing the symmetry condition, assuming that the centroid of the set lies at the origin, or replacing the volume functional by the surface area.2010 Mathematics Subject Classification. 52A40, 52A20, 52B11, 52C07.
Abstract. We consider the RMS-distance (sum of squared distances between pairs of points) under translation between two point sets in the plane. In the Hausdorff setup, each point is paired to its nearest neighbor in the other set. We develop algorithms for finding a local minimum in near-linear time in the line, and in nearly quadratic time in the plane. These improve substantially the worst-case behavior of the popular ICP heuristics for solving this problem. In the partial matching setup, each points in the smaller set is matched to a distinct point in the bigger set. Although the problem is not known to be polynomial, we establish several structural properties of the underlying subdivision of the plane and derive improved bounds on its complexity. In addition, we show how to compute a local minimum of the partial matching RMS-distance under translation, in polynomial time.
We consider the RMS distance (sum of squared distances between pairs of points) under translation between two point sets in the plane, in two different setups. In the partial-matching setup, each point in the smaller set is matched to a distinct point in the bigger set. Although the problem is not known to be polynomial, we establish several structural properties of the underlying subdivision of the plane and derive improved bounds on its complexity. These results lead to the best known algorithm for finding a translation for which the partial-matching RMS distance between the point sets is minimized. In addition, we show how to compute a local minimum of the partial-matching RMS distance under translation, in polynomial time. In the Hausdorff setup, each point is paired to its nearest neighbor in the other set. We develop algorithms for finding a local minimum of the Hausdorff RMS distance in nearly linear time on the line, and in nearly quadratic time in the plane. These improve substantially the worst-case behavior of the popular ICP heuristics for solving this problem.
We analyze a remarkable class of centrally symmetric polytopes, the Hansen polytopes of split graphs. We confirm Kalai's 3 d -conjecture for such polytopes (they all have at least 3 d nonempty faces) and show that the Hanner polytopes among them (which have exactly 3 d nonempty faces) correspond to threshold graphs. Our study produces a new family of Hansen polytopes that have only 3 d + 16 nonempty faces.
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