Reference for a particular Radon transform on non-positively curved spaces - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T06:39:21Z http://mathoverflow.net/feeds/question/31818 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/31818/reference-for-a-particular-radon-transform-on-non-positively-curved-spaces Reference for a particular Radon transform on non-positively curved spaces Benoît Kloeckner 2010-07-14T09:40:27Z 2011-06-30T20:22:13Z <p>Let me first recall that the classical Radon transform takes a (smooth compactly supported, say) function $f$ defined on $\mathbb{R}^n$ as an input, and gives as output the map $H\mapsto \int_H f$ for $H$ running over the set of affine hyperplanes. Radon's inversion theorem gives a formula to recover $f$ from its transform.</p> <p>I needed a similar result in a slightly different context in one of my project. We are on a simply connected, non-positively curved Riemannian manifold $X$ (or more generally a Hadamard space) and we have compactly supported smooth (positive if needed) functions $f$ and $g$. For all geodesic $\gamma\subset X$ (which is globally minimizing thanks to the curvature assumption, in particular it is convex), the metric projection $p_\gamma:X\to\gamma$ is well defined and $1$-Lipschitz. Let us call a <em>perpendicular</em> to $\gamma$ any level of $p_\gamma$. If we know that $\int_P f = \int_P g$ for all perpendicular $P$ of all geodesics, can we deduce that $f=g$?</p> <p>In fact, I managed to get a result along these lines which is sufficient for my needs; my question is: is this well-known? Does it have a name? Do you know a reference?</p> <p>Note that in the case of <code>$\mathbb{R}^n$</code> or of the real hyperbolic space, then perpendiculars are totally geodesic hyperplanes so that we are reduced to the usual Radon transform, and the book by Helgason contains more than I really need. But even when $X$ is the complex hyperbolic space, then the perpendiculars are not the subsets used in the usual Radon transform on symmetric spaces (note that they are not convex).</p> <p>Note also that the case of trees is an easy but nice combinatorial exercise; I could not find a reference either (I only found a paper studying the Radon transform defined using horospheres).</p> http://mathoverflow.net/questions/31818/reference-for-a-particular-radon-transform-on-non-positively-curved-spaces/31943#31943 Answer by m b for Reference for a particular Radon transform on non-positively curved spaces m b 2010-07-15T02:33:11Z 2010-07-15T02:33:11Z <p>If we work with Hadamard spaces, a perpendicular to a geodesic needn't be a geodesic, so I wonder what measure you use in the integral.</p>