The Optimal Multilinear Bohnenblust–Hille Constants: A Computational Solution for the Real Case

Optimal constants of classical inequalities in complex sequence spaces

“…(n) and n < ∞), for p 1 = • • • = p m = ∞, the optimal constants are "formally" known, using the algorithm developed in [9], but the time needed to run the algorithm is impeditive, with the current technology (see also [29]).…”

“…|T (e i1 , ..., e im )| T , where σ j are the identity maps, except for the case σ j (j) = 1 and σ j (1) = j. We thus have (29) C , where the notation is as in the Khinchine inequality for Steinhaus variables (Theorem 3.1). Note that when p = ∞, for reasons of symmetry, we recover (iii) for all bijections σ.…”

“…Recently, in [9], it was shown that the set C m.n is finite in the case of real scalars, and an algorithm was constructed, furnishing all the points of C m.n . So, with suitable computational assistance, the exact constants C R m (n) are determined (see also [29] for a successful implementation of the algorithm). The complex case remains open and seems to be even more difficult, because the geometry of the unit ball in the case of complex scalars is apparently "smoother" and the number of extreme points is quite likely infinite.…”