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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.

Informally, the underlying idea is providing a meaningful measure on the extent to which the points of $X$ are placed in $\mathbb{R}^2$ in such a away that any $\gamma$ passing through them has to curve away from any of its tangent vector at any point $\mathbf{p}\in\gamma$.

In other words $\gamma\in\Gamma(X)$ is the curve as flat as possible everywhere among the convex, simple and closed ones in $\mathbb{R}^2$ that pass through all points in $X$. Intuitively, the maximum curvature $\max_{\mathbf{p}\in\gamma}|\kappa_{\mathbf{n}}(\mathbf{p})|$ can also be viewed as a measure on the extent to which the points are far from being collinear in $\mathbb{R}^2$.

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.

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.

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.

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 $|\kappa_n(\mathbf{p})|$ 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.

Informally, the underlying idea is providing a meaningful measure on the extent to which the points of $X$ are placed in $\mathbb{R}^2$ in such a away that any $\gamma$ passing through them has to curve away from any of its tangent vector at any point $\mathbf{p}\in\gamma$.

In other words $\gamma\in\Gamma(X)$ is the curve as flat as possible everywhere among the convex, simple and closed ones in $\mathbb{R}^2$ that pass through all points in $X$. Intuitively, the maximum curvature $\max_{\mathbf{p}\in\gamma}|\kappa_n(\mathbf{p})|$ can also be viewed as a measure on the extent to which the points are far from being collinear in $\mathbb{R}^2$.

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.

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.

Informally, the underlying idea is providing a meaningful measure on the extent to which the points of $X$ are placed in $\mathbb{R}^2$ in such a away that any $\gamma$ passing through them has to curve away from any of its tangent vector at any point $\mathbf{p}\in\gamma$.

In other words $\gamma\in\Gamma(X)$ is the curve as flat as possible everywhere among the convex, simple and closed ones in $\mathbb{R}^2$ that pass through all points in $X$. Intuitively, the maximum curvature $\max_{\mathbf{p}\in\gamma}|\kappa_{\mathbf{n}}(\mathbf{p})|$ can also be viewed as a measure on the extent to which the points are far from being collinear in $\mathbb{R}^2$.

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.

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 $|\kappa_n(\mathbf{p})|$ 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.

Informally, the underlying idea is providing a meaningful measure on the extent to which the points of $X$ are placed in $\mathbb{R}^2$ in such a away that any $\gamma$ passing through them has to curve away from any of its tangent vector at any point $\mathbf{p}\in\gamma$.

In other words $\gamma\in\Gamma(X)$ is the curve as flat as possible everywhere among the convex, simple and closed ones in $\mathbb{R}^2$ that pass through all points in $X$. Intuitively, the maximum curvature $\max_{\mathbf{p}\in\gamma}|\kappa_n(\mathbf{p})|$ can also be viewed as a measure on the extent to which the points are far from being collinear in $\mathbb{R}^2$.

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

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 $|\kappa_n(\mathbf{p})|$ 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.

Informally, the underlying idea is providing a meaningful measure on the extent to which the points of $X$ are placed in $\mathbb{R}^2$ in such a away that any $\gamma$ passing through them has to curve away from any of its tangent vector at any point $\mathbf{p}\in\gamma$.

In other words $\gamma\in\Gamma(X)$ is the curve as flat as possible everywhere among the convex, simple and closed ones in $\mathbb{R}^2$ that pass through all points in $X$. Intuitively, the maximum curvature $\max_{\mathbf{p}\in\gamma}|\kappa_n(\mathbf{p})|$ can also be viewed as a measure on the extent to which the points are far from being collinear in $\mathbb{R}^2$.

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.