Hypersimplex frames beyond the simplex bound, and the exact table of quasimaximal projection constants up to N=11

Abstract

For a Banach space Y let λ(Y) be its absolute projection constant and λ_(K)(m)=sup{λ(Y):dim Y=m} the maximal projection constant in dimension m. Derk(e)gowska and Lewandowska recently showed that the m(m+1)/2 edge midpoints of a regular simplex in ℝᵐ form a Parseval frame of value 4(m-2)/(m+1), so that λ_(ℝ)(m) ≥ 4(m-2)/(m+1), and conjectured that this is sharp at m=6 and m=8. We contribute to the lower-bound side of this circle of problems. First, their frame is the k=2 member of a family indexed by k: the k-subsets of an n-set, projected onto the first nontrivial eigenspace of the Johnson scheme J(n,k) — the vertices of the hypersimplex Δ(n,k), and, as a frame, Datta and Oldroyd's k-angle tight frame. We evaluate the entrywise ℓ¹ mass of the corresponding projection in closed form and obtain λ_(ℝ)(n-1) ≥ v(n,k) for an explicit binomial sum v(n,k). Here k=2 returns the simplex bound, while k=3, whose value for n ≥ 9 is 6(n-4)(n-5)/(n(n-2)), exceeds it for every m=n-1 ≥ 13. In particular λ_(ℝ)(15) ≥ 99/28=3.5357… and λ_(ℝ)(16) ≥ 312/85=3.6705…, which exceed the values 3.5106 and 3.6157 that a recent numerical search reports in those dimensions. Second, we compute the quasimaximal relative projection constants μ_(ℝ)(m,N) exhaustively for 1 ≤ m ≤ N ≤ 11 in exact arithmetic. This reproduces every printed digit of Foucart and Skrzypek's table for N ≤ 10, supplies a closed algebraic form for each entry, and adds the row N=11, which their method could not reach; two of the new cells are exactly rational, μ_(ℝ)(5,11)=2 and μ_(ℝ)(6,11)=23/11, with exactly rational spectral projections. Every lower bound here is certified by an explicit integer matrix M and an integer D>0 with M=M^(T), M²=DM and tr M=mD, and every such certificate is machine-checked in Lean 4. We claim nothing about the upper half of the conjecture, and no asymptotic record: maxₖv(n,k) grows like √(m) with a constant below the one König, Lewis and Lin reached in 1983.

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