# Find a bounded function with a supporting point

Given, $g(Z)=\operatorname{tr}\phi(Z)$, where $\phi(Z)= Z^T\left( \operatorname{diag}(ZZ^T\mathbf{1}) - ZZ^T\right) Z$ where $Z$ is a real rectangular matrix with more rows than columns (tall and skinny) and Tr is the matrix trace:

I'd like to find a function $b(Z,U)$ such that $b(Z,U) \geq g(Z) , \forall Z \neq U$ and $b(Z,Z) = g(Z)$ only when $U=Z$. All matrices have real entries and matrices $U$ and $Z$ have the same dimensions.

Question: What would be an example function $b(.)$, that satisfies this condition?

One idea may be to find a matrix $M$ such that $M-\nabla^2{g(.)}$ is p.s.d and substituting the hessian matrix in the taylor expansion of $f(.)$ with M, but I wonder how I could find such an M.

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