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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 in the interval $(d,D)$ where $d>0$.

Does there exist a constant $c(d,D)>0$ (but independent of the metric) such that $\sup_x dist(x,-x)\geq c(d,D)$?

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 $D$.

Does there exist a constant $c(D)>0$ (but independent of the metric) such that $\sup_x dist(x,-x)\geq c(D)$?

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 in the interval $(d,D)$ where $d>0$.

Does there exist a constant $c(d,D)>0$ such that $\sup_x dist(x,-x)\geq c(d,D)$?

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 in the interval $(d,D)$ where $d>0$.

Does there exist a constant $c(d,D)>0$ (but independent of the metric) such that $\sup_x dist(x,-x)\geq c(d,D)$?

Let $g$ be a smooth Riemannian metric on $S^2$ which is invariant under the antipodal involution $x\mapsto -x$, has Gauss curvature at least $-1$ and diameter in the interval $(d,D)$ where $d>0$.

Does there exist a constant $c(d,D)>0$ such that $\sup_x dist(x,-x)\geq c(d,D)$?

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 in the interval $(d,D)$ where $d>0$.

Does there exist a constant $c(d,D)>0$ such that $\sup_x dist(x,-x)\geq c(d,D)$?