The lower bound for the number of facets of a k-neighborly d-polytope with d+3 vertices

We have found the minimal difference $\Delta(k) = \min\limits_P (f_{d-1}(P) - f_{0}(P))$betweenthenumberoffacetsandthenumberofverticesofa$k$-neighborly$d$-polytope$P$forthecase$f_{0}(P) = d+3$:$\Delta(2) = 4$,$\Delta(3) = 15$,and$\Delta(k) = 2 (k^2 - 1)$for$k \ge 4$.