  We investigate how much information about a convex body can be retrieved from
a finite number of its geometric moments. We give a sufficient condition for a
convex body to be uniquely determined by a finite number of its geometric
moments, and we show that among all convex bodies, those which are uniquely
determined by a finite number of moments form a dense set. Further, we derive a
stability result for convex bodies based on geometric moments. It turns out
that the stability result is improved considerably by using another set of
moments, namely Legendre moments. We present a reconstruction algorithm that
approximates a convex body using a finite number of its Legendre moments. The
consistency of the algorithm is established using the stability result for
Legendre moments. When only noisy measurements of Legendre moments are
available, the consistency of the algorithm is established under certain
assumptions on the variance of the noise variables.
