Geometry

Sometime in late 2012 or early 2013, Joel Tropp, Martin Lotz, Dennis Amelunxen, and I were all discussing some integral-geometric problem or another, and the topic of projections of convex cones came up. One of us made the observation that taking the linear image of a subspace in general position yields one of two results: Either the subspace is “small enough” for the linear map, and so remains a linear space of the same dimension; or The subspace is “too large” for the linear map, and so it fills the entire space. This elementary observation, combined with the spherical Hadwiger formula, yields a startling result: many geometric quantities related to convex cones are, on average, precisely preserved under linear maps. Based on the notation we had on the chalkboard at the time, we called this result the TQC lemma. ...