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Authors
Janos Kollar
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The aim of this note is to prove an analog of the flattening decomposition theorem for reflexive hulls. The main applications are: the construction of the moduli space of varieties of general type, improved flatness conditions and criteria for simultaneous normalizations.
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This paper presents a new algorithm for the convex hull problem, which is based on a reduction to a combinatorial decision problem POLYTOPE-COMPLETENESS-COMBINATORIAL, 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 POLYTOPE-COMPLETENESS-COMBINATORIAL has a certificate that can be checked in polynomial time (if integrity of the input is guaranteed).
We give conditions characterizing holomorphic and meromorphic functions in the unit disk of the complex plane in terms of certain weak forms of the maximum principle. Our work is directly inspired by recent results of John Wermer, and by the theory of the projective hull of a compact subset of complex projective space developed by Reese Harvey and Blaine Lawson.
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Given a set $S$ of $n$ points in the Euclidean plane, the two-center problem is to find two congruent disks of smallest radius whose union covers all points of $S$. Previously, Eppstein [SODA97] gave a randomized algorithm of $O(nlog^2n)$ expected time and Chan [CGTA99] presented a deterministic algorithm of $O(nlog^2 nlog^2log n)$ time. In this paper, we propose an $O(nlog^2 n)$ time deterministic algorithm, which improves Chans deterministic algorithm and matches the randomized bound of Eppstein. If $S$ is in convex position, then we solve the problem in $O(nlog nloglog n)$ deterministic time. Our results rely on new techniques for dynamically maintaining circular hulls under point insertions and deletions, which are of independent interest.
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