Pure Maths Colloquium: The best path joining two segments on the line
29 May 2006 00:00 in CM221
"Let S be the set of all compact intervals on the real line having more than one point. It is a surface which is not canonically embedded in Euclidean space. We present two metrics on S and study their geodesics. The first one, which does not discriminate size (each segment takes itself as yardstick to measure the size of the neighbouring segments and the distance to them), is a model of the hyperbolic plane. We also comment on more general situations. The second one, which takes into account the individual motions of the interior points of the segments is a simple two dimensional example motivating the canonical metric on the Frechet space of all embeddings of a fixed compact manifold in Euclidean space. "
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