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December 30th, 2017, 05:09 PM  #1 
Senior Member Joined: Oct 2013 From: New York, USA Posts: 622 Thanks: 85  Proving The Maximum Area of a Shape Regardless of How Many Sides It Has
The greatest area of a quadrilateral given a fixed perimeter is a square. This seems to be true for an equilateral triangle. Is there a proof that given a fixed perimeter and fixed amount of sides, the area will be maximized when all the sides are equal regardless of how many sides there are?

December 30th, 2017, 06:00 PM  #2 
Senior Member Joined: Sep 2016 From: USA Posts: 520 Thanks: 293 Math Focus: Dynamical systems, analytic function theory, numerics 
I'm a bit short on time atm but here is the idea. Suppose a $n$polygon is specified as a list of $n+1$ vertices, $\{(x_0,y_0),\dotsc,(x_n,y_n)\}$ and let $\gamma_k$ denote the line segment between the $(k1)^{\rm st}$ and $k^{\rm th}$ vertex. Then the area is given explicitly by the line integral \[\sum_{k=1}^n \int_{\gamma_k} x \ dy \] Now, regard this as a function of $2(n+1)$many variables and maximize it using standard multivariable optimization techniques. 
December 31st, 2017, 01:46 PM  #3 
Global Moderator Joined: May 2007 Posts: 6,641 Thanks: 625 
For any figure more than 3 sides, it is important to include a requirement that all angles be equal. Example for 4 sides: a rhombus has all sides equal, but it can be squeezed to an area as close to 0 as one wants. 
January 4th, 2018, 05:19 PM  #4 
Senior Member Joined: Feb 2016 From: Australia Posts: 1,737 Thanks: 606 Math Focus: Yet to find out.  

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area, maximum, proving, shape, sides 
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