Grothendieck constant is norm of Strassen matrix multiplication tensor

We show that two important quantities from two disparate areas of complexity theory — Strassen’s exponent of matrix multiplication ω and Grothendieck’s constant KG — are intimately related. They are different measures of size for the same underlying object — the matrix multiplication tensor, i.e., the 3-tensor or bilinear operator μl,m,n : Fl×m × Fm×n → Fl×n, (A,B) 7→ AB defined by matrix-matrix product over F = R or C. It is well-known that Strassen’s exponent of matrix multiplication is the greatest lower bound on (the log of) a tensor rank of μl,m,n. We will show that Grothendieck’s constant is the least upper bound on a tensor norm of μl,m,n, taken over all l,m, n ∈ N. Aside from relating the two celebrated quantities, this insight allows us to rewrite Grothendieck’s inequality as a norm inequality ‖μl,m,n‖1,2,∞ = max X,Y,M 6=0 | tr(XMY )| ‖X‖1,2‖Y ‖2,∞‖M‖∞,1 ≤ KG, and thereby allows a natural generalization to arbitrary p, q, r, 1 ≤ p, q, r ≤ ∞. We show that the following generalization is locally sharp: ‖μl,m,n‖p,q,r = max X,Y,M 6=0 | tr(XMY )| ‖X‖p,q‖Y ‖q,r‖M‖r,p ≤ KG · l ·m · n, and conjecture that Grothendieck’s inequality is unique: (p, q, r) = (1, 2,∞) is the only choice for which ‖μl,m,n‖p,q,r is uniformly bounded by a constant. We establish two-thirds of this conjecture: uniform boundedness of ‖μl,m,n‖p,q,r over all l,m, n necessarily implies that min{p, q, r} = 1 and max{p, q, r} =∞.