EDIT: Let $g$ be a smooth Riemannian metric on $S^2$ which is invariant under the antipodal involution $x\mapsto -x$. Assume it has Gauss curvature at least $-1$ and diameter $D$. 



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> Does there exist a constant $c(D)>0$ (but independent of the metric) such that $\sup\_x dist(x,-x)\geq c(D)$? 
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Riemannian metrics on 2-sphere invariant under antipodal involution
The answer is "yes".


Let $g$ be the induced Riemannian metric on $\mathbb R\mathrm P^2=\mathbb S^2/\mathbb Z\_2$.
Denote by $\ell$ the systole of $(\mathbb R\mathrm P^2,g)$ and let $\gamma$ be the corresponding closed geodesic.
Let us cut $(\mathbb R\mathrm P^2,g)$ along $\gamma$. 
We obtain a disc $\Delta$ with diameter is at least $D/2$ and its boundary has length $2\cdot\ell$.


Note that we can assume that $\ell<D/100$.
Therefore there is a point $x$ in $\Delta$ that lies on distance at least $D/10$ from the boundary.
It remains to note that the distance from $x$ to its antipodal point is more than $D/10$.


