On the reconstruction of convex sets from random normal measurements

Abstract: We study the problem of reconstructing a convex body using only a finite number of measurements of outer normal vectors. More precisely, we suppose that the normal vectors are measured at independent random locations uniformly distributed along the boundary of our convex set. Given a desired Hausdorff error eta, we provide an upper bounds on the number of probes that one has to perform in order to obtain an eta-approximation of this convex set with high probability. Our result rely on the stability theory related to Minkowski's theorem.