Jan Vybíral

An explicit counterexample to the Hinrichs--Vybiral conjecture

Conjecture~2 of Hinrichs and Vyb\'iral asserts that every continuous, nonnegative, positive definite function $f$ on $\R^d$, normalized by $f(0)=1$, satisfies $[f(x_j-x_k)]_{j,k=1}^n\succeq \one\one^T/n$. We give an explicit trigonometric polynomial on $\R^7$ and eight points for which this inequality fails. All hypoth...