ℓ^p variant for 2 < p < ∞

Determine the sharp value or asymptotic behavior of β_{n,p} = max over A ∈ ℝ^{n×n} with unit ℓ2-norm rows of (1/2^n) ∑_{x∈{−1,1}^n} ||Ax||_p for 2 < p < ∞, and identify the extremal matrices that achieve it.

Background

The paper fully resolves the cases p = 1 and p = 2 and establishes an upper bound of order c√p * n^{1/p} for p > 2. However, the exact behavior for intermediate p remains unknown.

Thus the open problem is to determine tight bounds (or exact values) and extremizers for the ℓ^p version of the bad science matrix problem when 2 < p < ∞.