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We prove that the geometric covariogram determines (up to translation and reflection), among all convex bodies, any plane convex body which is C 2 and has positive curvature everywhere. This gives a partial answer to a problem posed by G. Matheron.

Abstract. There are sequences of directions such that, given any compact set K ⊂ R n , the sequence of iterated Steiner symmetrals of K in these directions converges to a ball. However examples show that Steiner symmetrization along a sequence of directions whose differences are square summable does not generally converge. (Note that this may happen even with sequences of directions which are dense in S n−1 .) Here we show that such sequences converge in shape. The limit need not be an ellipsoid or even a convex set.We also deal with uniformly distributed sequences of directions, and with a recent result of Klain on Steiner symmetrization along sequences chosen from a finite set of directions.

The authors complement previous work of Lorentz (1949) on the determination of plane measurable sets from two projections. They give three different characterisations of uniquely determined sets and show a reconstruction formula for such sets.

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