Reconstruction of  convex sets  from support- or  brightness functions

We discuss how a convex compact subset of $n$-dimensional Euclidean space can be reconstructed from finitely many noisy measurements. The measurements are either brightness function values (shadow areas of the set after orthogonally projecting on hyperplanes), or support function values (distances of touching hyperplanes from the origin).

In the talk I will show explicit reconstruction algorithms that produce polytopes as output. Under appropriate assumptions, the algorithms are strongly consistent, meaning that the output converges to the true set in an appropriate metric, when the number of measurements converges to infinity. These results depend on uniform laws of large numbers, derived in the theory of empirical processes, and this connection will be sketched in the talk as well.