1

Here "smallest" refers to the area. An upper bound would be a circle with diameter $p/2$ and area $p^2/16$. A lower bound would be the envelope of all isoperimetric ellipses centered at origin with their long axis along x-axis. Its parametric function is derived here but I don't think it's trivial to find its area. And a square won't fit into this shape.

EDIT: "fit into" allows translation, rotation, and mirroring

The shape is no smaller than yellow, but no larger than blue

arax
  • 2,815
  • It's pretty obvious that the smallest shape is the circle of radius $\frac p4$. You can show this by first noting that every concave shape can fit inside a convex shape of smaller perimeter, and then considering the properties of convex shapes. – Rushabh Mehta Nov 02 '19 at 04:53
  • 1
    @DonThousand- Are you assuming that you don't get to control the orientation of the shape? Obviously the shape must have a maximum diameter of $p/2$. but any curve approaching this maximum diameter is not going to need to extend anywhere near $\pi/4$ in the perpendicular direction. – Paul Sinclair Nov 02 '19 at 16:32

0 Answers0