A Beck - Fiala-type Theorem for Euclidean Norms

The inequality in (a) is naturally the best possible. Theorem l(a) is connected with questions such as the Beck-Fiala theorem or the Koml6s conjecture (see [2J, [3J and [7]). Combinatorial motivations are presented exhaustively in [8J (see also [6]). In a slightly different form, Theorem l(a) was used in [IJ in the proof that nuclear Frechet spaces satisfy the Levy-Steinitz theorem on rearrangement of series (or, more precisely, that a nuclear operator acting between Hilbert spaces is a Steinitz operator). If D is the euclidean unit ball in [Rn, the right-hand side in (a) is equal to Vii; this special case was obtained by S. Sevastyanov [5J and, independently, by I. Barany (unpublished). Theorem l(b) is connected with the so-called Steinitz lemma (see [4], especially Theorem 2). It confirms the hy,Pothesis that the Steinitz constant of the n-dimensional euclidean space is of order Vn as n-H~) (or, more generally, that Hilbert-Schmidt operators are Steinitz operators); see [1, Remarks 8 and 7). Another consequence of Theorem l(b) is Theorem 2, given at the end of the paper. The author is indebted to I. Barany and S. Sevastyanov for their comments on the subject.