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Is it true that for centrally symmetric(symmetric with respect to the origin), strictly convex closed curves, if the support function at a point is minimum then curvature is minimum at that point and if the support function is maximum at a point then the curvature is maximum at that point.

Edit: Graph of the intersection body suggested by user8268 for $\varepsilon=1$

 Graph of the intersection body suggested by user8268  for $\varepsilon=1$

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No. Take e.g. the ellipses $((x+\epsilon)/2)^2+y^2\leq 1$ and $((x-\epsilon)/2)^2+y^2\leq 1$ (for some small $\epsilon$). Their intersection is bounded by a convex symmetric curve with $2$ "corners". At the corners the curvature is infinite. (if you prefer a smooth curve, take this one and smooth out the corners so that the curvature there stays very large).

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