Jan Boman: The Interior Problem for the Radon transform 2016 - 2026, a story with several surprises

Abstract: Let K be a compact, convex subset of the plane and K_0 a closed subset of the interior of K. The Interior Problem for the 2-dimensional Radon transform asks whether one can recover the restriction to K_0 of a function f(x) supported in K from knowledge of the Radon transform R(f(L)) for lines L that intersect K_0. In Frank Natterer's book from 1986, The mathematics of Computerized Tomography, it is proved that the answer is NO in the case when K is a disk: radial functions are constructed with support equal to K (origin-centered) for which the Radon transform vanishes for all lines that intersect K_0. In 2016 I wanted to prove the same in the general case by choosing a convex set D with boundary \partial D contained in the interior of K\setminus K_0 and construct a distribution f \ne 0 with Radon transform supported in the set of tangent lines to \partial D. This idea failed, because such distributions exist only if \partial D is an ellipse (first surprise). However, that fact --- also true in higher dimensions --- turned out to give a new proof for a recent result on another problem (second surprise) and therefore led to four publications. Last year I published an Extended abstract containing a non-constructive existence proof of a non-trivial function supported in K with Radon transform vanishing for all lines that intersect K_0. This was a step in the desired direction, because examples show that almost all functions with those properties should be different from zero somewhere in K_0. However, shortly before I got the proofs I had observed that all such functions must have support that covers K_0 (third surprise). So the desired extension of the 40 years old theorem was proved.