The curse of the Euclidean metric: corner.mimuw.edu.pl/?p=1073

Krzysztof Fleszar posts about a big difficulty with algorithms for problems like Euclidean shortest paths where the answer is a sum of distances: we don't know how to compare two solutions efficiently.

Krzysztof's SODA 2019 paper on approximate TSP of hyperplanes (find a short tour that touches each given hyperplane) is epubs.siam.org/doi/10.1137/1.9 — fortunately in this case it's possible to use approximate numerical comparisons.

Paul Erdős died in 1996, but his most recent paper is from 2015, nearly 20 years later! There's a writeup at simonsfoundation.org/2015/12/1; the paper itself is math.ucsd.edu/~ronspubs/pre_tr

It's about Egyptian fractions – representations of rationals as sums of distinct unit fractions – and is motivated by the conjecture that it's always possible for all denominators to be semiprime. That's still open, but they prove that every integer has a representation with all denominators products of three primes.

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