Ian Doust University of New South Wales
Balanced matrices and functions There is an interesting family of norms on ^{n} given by
where the maximum is taken over all k element subsets of {1,…,n}. Let A be an n×n matrix with rows r_{1},…,r_{n}
and columns c_{1},…c_{n}. We say that A is kbalanced if ∥r_{i}∥_{1,k} = R for all i and ∥c_{j}∥_{1,k} = C for all j. A little
experimentation shows that R and C can be different. (The first interesting case occurs with n = 4
and k = 3.) Finding the optimal inequalities relating R and C has proven to be a challenge and
there are still many open problems. The concept of a balanced function is defined analogously,
now integrating of sets of a fixed measure. Our state of knowledge here is much more limited,
especially if one restricts one’s attention to continuous functions. The big open question is whether a
continuous balanced function on [0,1] × [0,1] exists which has a nontrivial ratio R∕C? This is
joint hobby mathematics with Richard Aron (Kent State) and Nigel Kalton (Columbia, Missouri).
