Geometric Discrepancy Revisited

Bertrand’s Paradox Revisited: More Lessons about that Ambiguous Word, Random

The Bertrand paradox question is: “Consider a unit-radius circle for which the length of a side of an inscribed equilateral triangle equals 3 . Determine the probability that the length of a ‘random’ chord of a unit-radius circle has length greater than 3 .” Bertrand derived three different ‘correct’ answers, the correctness depending on interpretation of the word, random. Here we employ geomet...

Chebyshev Sets, Klee Sets, and Chebyshev Centers with respect to Bregman Distances: Recent Results and Open Problems

In Euclidean spaces, the geometric notions of nearest-points map, farthestpoints map, Chebyshev set, Klee set, and Chebyshev center are well known and well understood. Since early works going back to the 1930s, tremendous theoretical progress has been made, mostly by extending classical results from Euclidean space to Banach space settings. In all these results, the distance between points is i...

The matrix arithmetic-geometric mean inequality revisited

Bracketing numbers for axis-parallel boxes and applications to geometric discrepancy

In the first part of this paper we derive lower bounds and constructive upper bounds for the bracketing numbers of anchored and unanchored axis-parallel boxes in the d-dimensional unit cube. In the second part we apply these results to geometric discrepancy. We derive upper bounds for the inverse of the star and the extreme discrepancy with explicitly given small constants and an optimal depend...

Using low-discrepancy sequences and the Crofton formula to compute surface areas of geometric models

The surface area of a geometric model, like its volume, is an important integral property that needs to be evaluated frequently and accurately in practice. In this paper we present a new quasi-Monte