We are given a set $X$ of $n\ge 3$ points in $\mathbb{R}^2$, belonging to the boundary of the convex hull of $X$ itself. Let $\Gamma(X)$ be the set of all *convex, simple closed* curves in $\mathbb{R}^2$ passing through all points in $X$ such that, at any point $\mathbf{p}\in\gamma$, there exists a normal vector $\mathbf{n}\_{\mathbf{p}}$ and a tangent vector $\mathbf{v}\_{\mathbf{p}}$ lying on $\mathbb{R}^2$. Let $\kappa\_{\mathbf{n}}(\mathbf{p})$ the curvature of $\gamma$ at $\mathbf{p}$ in the direction of $\mathbf{v}\_{\mathbf{p}}$.

I am looking for the name of the curve(s) $\gamma\in\Gamma(X)$ that ***minimizes*** the ***maximum*** curvature absolute value $|\kappa\_{\mathbf{n}}(\mathbf{p})|$ *over all* points $\mathbf{p}\in\gamma$. I view it as a natural mathematical concept; therefore, I expect it is already mentioned in the related literature (although I could not find any reference yet).

*Besides the name (if it exists), I would be glad to have any reference about this topic.*

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***Edit: After the discussion with SaúlRM and its answer, I added the constraint that $\gamma$ is convex.***

***Edit: After the comment of Yaakov Baruch, I added the constraint that all points of $X$ belong to the boundary of the convex hull of $X$ itself.***