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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

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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

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  • $\begingroup$ 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

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