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If $\vec{y}_1$ and $\vec{y}_2$ are the gradients of a differentiable convex function $f(x)$ at points $\vec{x}_1$ and $\vec{x}_2$ does there exist a $\nabla f^{-1}( \alpha \vec{y}_1 + (1-\alpha)\vec{y}_2)$ in the domain of $f.$

The forward image of the domain mapped by the gradient is the domain of the Legendre conjugate so I think the above is true. If you could point me to a proof that will be very helpful.

I can see that the Legendre conjugate is convex because it is a supremum of affine functions, but am looking for a more direct intuition / construction for the forward image of the domain mapped by the gradient. What is bothering me is what happens when the supremum that is used to define the conjugate is not achieved. The value of the conjugate is set to $\infty$ but is there a way to show that the set over which the supremum is achieved is convex (even if $f$ is unbounded below).

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