Suppose $K$ is a centrally symmetric, strictly convex body in $\mathbb{R}^2$. Let denote the curvature and the support function of $\partial K$, boundary of $K$, respectively with $\kappa$ and $s$. If $m\le\frac{\kappa}{s^3}(K)\le M$ for some positive numbers $m$ and $M$, does it mean there are ellipsoids $E_1$ and $E_2$ such that $E_1\subseteq K\subseteq E_2$ and $$\frac{\kappa}{s^3}({E_1})=m,~~~ $\frac{\kappa}{s^3}({E_1})=M,~~~ \frac{\kappa}{s^3}(E_2)=M frac{\kappa}{s^3}(E_2)=m ~~~~? $$
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minimal maximal ellipsoidsSuppose $K$ is a centrally symmetric, strictly convex body in $\mathbb{R}^2$. Let denote the curvature and the support function of $\partial K$, boundary of $K$, respectively with $\kappa$ and $s$. If $m\le\frac{\kappa}{s^3}(K)\le M$ for some positive numbers $m$ and $M$, does it mean there are ellipsoids $E_1$ and $E_2$ such that $E_1\subseteq K\subseteq E_2$ and $$\frac{\kappa}{s^3}({E_1})=m,~~~ \frac{\kappa}{s^3}(E_2)=M ~~~~? $$
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