We say a rectangle has orientation $\theta$ if the vector from its center to the middle of its shortest side (parallel to the longest side) has some angle $\theta$ with X axis.

Consider a planar convex region $C$ fixed in $\mathbb{R}^2$. Let us draw $R_{1}$, the smallest rectangle containing $C$ and call the orientation of $R_{1}$, $\theta_1$. Let us also draw the rectangle $R_{2}$, the largest rectangle contained within $C$ and let its orientation be $\theta_2$.

Example illustration

Question: Which planar convex region $R$ maximizes $|\theta_1-\theta_2|$ ?

Note 1: If either rectangle is a square, the $\theta$ values won't be unique and we take the smallest value of $|\theta_1-\theta_2|$ as the orientation difference.

Note 2: The same question can be asked with area replaced with perimeter.

Related: bounds on largest internal rectangle area and considering a simple case: right triangles

We say a rectangle has orientation $\theta$ if the vector from its center to the middle of its shortest side (parallel to the longest side) has some angle $\theta$ with X axis.

Consider a planar convex region $C$ fixed in $\mathbb{R}^2$. Let us draw $R_{1}$, the smallest rectangle containing $C$ and call the orientation of $R_{1}$, $\theta_1$. Let us also draw the rectangle $R_{2}$, the largest rectangle contained within $C$ and let its orientation be $\theta_2$.

Question: Which planar convex region $R$ maximizes $|\theta_1-\theta_2|$ ?

Note 1: If either rectangle is a square, the $\theta$ values won't be unique and we take the smallest value of $|\theta_1-\theta_2|$ as the orientation difference.

Note 2: The same question can be asked with area replaced with perimeter.

Related: bounds on largest internal rectangle area and considering a simple case: right triangles

We say a rectangle has orientation $\theta$ if when centered at zero the vector from $(0,0)$ to the middle of the shortest side (parallel to the longest side) has some angle $\theta$ with X axis.

Consider a planar convex region $C$ in $\mathbb{R}^2$. Let $R_{1}$ be the smallest rectangle containing it. Let the orientation of $R_{1}$ be $\theta_1$. Let the rectangle $R_{2}$ be the largest rectangle contained in $C$ with orientation $\theta_2$.

Example illustration

Question: Which planar convex region $R$ maximizes $|\theta_1-\theta_2|$ ?

Note 1: If either rectangle is a square, the $\theta$ values won't be unique and we take the smallest value of $|\theta_1-\theta_2|$ as the orientation difference.

Note 2: The same question can be asked with area replaced with perimeter.

Related: bounds on largest internal rectangle area and considering a simple case: right triangles

We say a rectangle has orientation $\theta$ if the vector from its center to the middle of its shortest side (parallel to the longest side) has some angle $\theta$ with X axis.

Consider a planar convex region $C$ fixed in $\mathbb{R}^2$. Let us draw $R_{1}$, the smallest rectangle containing $C$ and call the orientation of $R_{1}$, $\theta_1$. Let us also draw the rectangle $R_{2}$, the largest rectangle contained within $C$ and let its orientation be $\theta_2$.

Example illustration

Question: Which planar convex region $R$ maximizes $|\theta_1-\theta_2|$ ?

Note 1: If either rectangle is a square, the $\theta$ values won't be unique and we take the smallest value of $|\theta_1-\theta_2|$ as the orientation difference.

Note 2: The same question can be asked with area replaced with perimeter.

Related: bounds on largest internal rectangle area and considering a simple case: right triangles

Consider a planar convex region $C$ in $\mathbb{R}^2$.

We say a rectangle has orientation $\theta$ if when centered at zero the vector from $(0,0)$ to the middle of the shortest side (parallel to the longest side) has some angle $\theta$.

Let the circumscribed rectangle $R_{1}$ be the smallest rectangle containing it, with orientation $\theta_1$.

  Let the inscribed rectangle $R_{2}$ be the largest rectangle contained in it, with orientation $\theta_2$.

Example illustration

Which planar convex region $R$ maximizes $|\theta_1-\theta_2|$ ?

Note 1: The same question can be asked with area replaced with perimeter.

Note 2: $\theta$ may not be unique in degenerate square case for either $R$.

Related: bounds on largest internal rectangle area and considering a simple case: right triangles

We say a rectangle has orientation $\theta$ if when centered at zero the vector from $(0,0)$ to the middle of the shortest side (parallel to the longest side) has some angle $\theta$ with X axis.

Consider a planar convex region $C$ in $\mathbb{R}^2$. Let $R_{1}$ be the smallest rectangle containing it. Let the orientation of $R_{1}$ be $\theta_1$. Let the rectangle $R_{2}$ be the largest rectangle contained in $C$ with orientation $\theta_2$.

Example illustration

Question: Which planar convex region $R$ maximizes $|\theta_1-\theta_2|$ ?

Note 1: If either rectangle is a square, the $\theta$ values won't be unique and we take the smallest value of $|\theta_1-\theta_2|$ as the orientation difference.

Note 2: The same question can be asked with area replaced with perimeter.

Related: bounds on largest internal rectangle area and considering a simple case: right triangles