For a fixed area, what kind of polygon that would obtain the maximum area of its convex hull.

For e.g. area 1 square, since it’s already convex, so the convex hull having area 1 too. Would there be any shaped polygon having area 1 but with its convex hull having area more than 1? If so what would be the shape you get the maximum convex hull area


You can construct a shape with limited area of any convex hull. Here's a simple solution. Points are $(1,0),(s+1,0),(1,\frac{1}{2s}),\left(1,1\right),\left(\frac{1}{2s},1\right),\left(0,s+1\right),\left(0,1\right)$

This is an octagon with area $1$ and convex hull $(s+1)^2/2-\frac12$

Another (easier) solution with a quadrilateral is $\left(0,0\right),\left(a,0\right),\left(\frac{1}{a},\frac{1}{a}\right),\left(0,a\right)$

desmos graph

  • $\begingroup$ Thank you, adding images would be amazing! $\endgroup$ – peng yu Jul 28 '20 at 3:43
  • $\begingroup$ I added the picture and both solutions to the desmos link $\endgroup$ – Tbw Jul 28 '20 at 3:58

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.