Does $zx^T - x y^T - y z^T \ge 0$ hold?

Question

Suppose $x,y,z$ are three row vectors of the same length, and they satisfy :

(1) each element in $x,y,z$ is between 0 and 1

(2) $\|x\|_{\infty} \le 1, \quad \|y\|_{\infty} \le 1, \quad \|z\|_{\infty} \le 1$

(NOTE: $\|x\|_{\infty}$ means the row sum of $x$)

(3) each column of the matrix $\left[ \begin{matrix} x \\ y \\ z\end{matrix} \right]$ has only two nonzero elements.

My question is: Does this inequality hold?

$zx^T - x y^T - y z^T \ge 0$

Answer 1

No, a simple counter-example is $$ \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} a & a & 0 \\ a & 0 & a \\ 0 & a & a \end{pmatrix} $$ for any $0 < a \le 1/2$. Each row sum is $2a$ and $$ zx^T - x y^T - y z^T = a^2 - a^2 - a^2 = -a^2 < 0 \, . $$