Symmetry and convexity · Question 5

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If we replace condition in the theorem with $\nabla^2 W(x) \geq F(x) \, Id$, can we obtain better results?

## Question context

\begin{theorem}[Caffarelli]
 	Let $T = \nabla \Phi: \mu \to \nu$ be a Brenier map, where $\mu$  and $\nu$ are probability measures with densities $e^{-V(x)}$ and $e^{-W(x)}$. Suppose that
 $$ 
 	\nabla^2 W(x) \geq K \, Id,
 $$	
for some constant $K$. Then for any $e \in S^{n-1}$
 	$$
 	\sup\limits_{ x \in \mathbb{R}^n } \Phi^2_{ e e } \leq \frac1K \sup\limits_{x \in \mathbb{R}^n}  V_{ee},
 	$$
 	where $\Phi_{e}$ denotes directional derivative.
 \end{theorem}

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