In the attempt of solving the isoperimetric problem in 1838 Steiner missed the important point of the existence of the solution. Various authors found alternative proofs or filled the gap. W. Gross belongs to the second group (1917). He constructed, given a convex body K, a sequence of directions {u_n} such that iterated Steiner symmetrals taken in that directions minimize the perimeter and converge, in the Hausdorff distance, to the ball K^* centered at the origin and having the same volume as K.

Peter Mani (1986) was the first to understand that the convergence of iterated Steiner symmetrizations of a convex body K to K^* holds almost surely and conjectured that this happens also if we consider successive Steiner symmetrizations of a compact set. The conjecture was confirmed in 2006 by van Schaftingen. I provided in 2013 a different proof.

I conjectured in 2009 that density alone of the sequence {u_n} is sufficient for the convergence of the iterated Steiner symmetrization. P. Gronchi disproved the conjecture in 2010.

Bianchi, Klain, Lutwak, Yang and Zhang (2011) proved however the following result:

Theorem. If K is a convex body and U is a countable and dense set of directions, then it can be ordered in a sequence {u_n} such that the successive Steiner iterations of K in that directions converge in the Hausdorff metric to the ball K^*.

I improved recently this result in two directions (2015, visible on line). On one hand the seed K of the iteration is allowed to be compact rather than just convex, and moreover there exists a universal ordering of U which works for every seed.