Economical Delone Sets for Approximating Convex Bodies

Convex bodies are ubiquitous in computational geometry and optimization theory. The high combinatorial complexity of multidimensional convex polytopes has motivated the development of algorithms and data structures for approximate representations. This paper demonstrates an intriguing connection between convex approximation and the classical concept of Delone sets from the theory of metric spaces. It shows that with the help of a classical structure from convexity theory, called a Macbeath region, it is possible to construct an epsilon-approximation of any convex body as the union of O(1/epsilon^{(d-1)/2}) ellipsoids, where the center points of these ellipsoids form a Delone set in the Hilbert metric associated with the convex body. Furthermore, a hierarchy of such approximations yields a data structure that answers epsilon-approximate polytope membership queries in O(log (1/epsilon)) time. This matches the best asymptotic results for this problem, by a data structure that both is simpler and arguably more elegant

Claude Ambrose Rogers. 1 November 1920 — 5 December 2005

Claude Ambrose Rogers and his identical twin brother, Stephen Clifford, were born in Cambridge in 1920 and came from a long scientific heritage. Their great-great-grandfather, Davies Gilbert, was President of the Royal Society from 1827 to 1830; their father was a Fellow of the Society and distinguished for his work in tropical medicine. After attending boarding school at Berkhamsted with his twin brother from the age of 8 years, Ambrose, who had developed very different scientific interests from those of his father, entered University College London in 1938 to study mathematics. He completed the course in 1940 and graduated in 1941 with first-class honours, by which time the UK had been at war with Germany for two years. He joined the Applied Ballistics Branch of the Ministry of Supply in 1940, where he worked until 1945, apparently on calculations using radar data to direct anti-aircraft fire. However, this did not lead to research interests in applied mathematics, but rather to several areas of pure mathematics. Ambrose's PhD research was at Birkbeck College, London, under the supervision of L. S. Bosanquet and R. G. Cooke, his first paper being on the subject of geometry of numbers. Later, Rogers became known for his very wide interests in mathematics, including not only geometry of numbers but also Hausdorff measures, convexity and analytic sets, as described in this memoir. Ambrose was married in 1952 to Joan North, and they had two daughters, Jane and Petra, to form a happy family