A reverse isoperimetric inequality for convex shapes with inclusion constraint

Abstract

The convex shape contained in a disk having prescribed area and maximal perimeter is completely characterized in terms of the area fraction. The solution is always a polygon having all but one sides equal. The lengths of the sides are characterized through explicit equations. The case of more general containing shapes is also discussed from both theoretical and numerical perspectives.

Figures (5)

• Figure 1: An optimal $n$-gon does not have a free vertex inside the container $D$.
• Figure 2: Values of $\lambda$ in terms of $m$ and $\theta$ given in \ref{['eq:lambdas']} for $3 \leq m \leq 10$.
• Figure 3: Maximal perimeter convex sets contained in the unit disk $D$ for different area fractions $A \in (0,|D|)$. Changes in the number of edges occur when $A$ is the area of a regular $n$-gon inscribed in $D$.
• Figure 4: Numerical results for maximization of the perimeter among polygons with $10,30$ and $50$ sides inscribed in $\Omega_1$ (with radial function given by \ref{['eq:rad_general_1']}) with given area fraction $A$.
• Figure 5: Numerical results for maximization of the perimeter among polygons with $10,30$ and $50$ sides inscribed in $\Omega_1$ (with radial function given by \ref{['eq:rad_general_2']}) with given area fraction $A$.

Theorems & Definitions (11)

• Theorem 1
• Theorem 2
• Proposition 3
• Proposition 4
• Proposition 5
• Theorem 6
• Theorem 7
• Theorem 8
• Remark 9
• Theorem 10