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Enrico Au-Yeung (DePaul University)


Analysis meets Graph theory: conical Radon transform and red, blue, purple triangles


The following two problems, one from Analysis, and one from Intuitive Geometry, how are these two problems connected?

We consider the inversion of a conical Radon transform that maps a function defined in 3-dimensional space to its integrals over a special family of cones. Recovering a function from integrals over cones depends on the collection of cones. There are over a dozen ways of specifying a family of cones; by the location of the vertex of each cone, the axis of symmetry, and the angle the vertex of the cone makes with the axis. The problem becomes even more interesting if we restrict the collection to a finite set of cones.

Next, consider a problem that arise in Intuitive Geometry. A multicolored triangle is one where each vertex is in a different colour, i.e. the vertices are red, blue, and purple. Suppose there is a set of n points in the plane, where 180 of them are red, 180 are blue, and 180 are purple. Then, it is possible to select a subset of 15 points from each color (red, blue, and purple), so that if we form all multicolored triangles from these 45 points, then these triangles have a non-empty intersection.