## Shrinkable polygons

Here’s a nice problem inspired from a post on MathOverflow: link

We call a polygon *shrinkable* if any scaling of itself with a factor can be translated into itself. Characterize all shrinkable polygons.

It is easy to see that any star-convex polygon is shrinkable. Pick the point in the definition of star-convex, and any contraction of the polygon by a homothety of center lies inside the polygon.

## Numerical method – minimizing eigenvalues on polygons

I will present here an algorithm to find numerically the polygon with sides which minimizes the -th eigenvalue of the Dirichlet Laplacian with a volume constraint.

The first question is: how do we calculate the eigenvalues of a polygon? I adapted a variant of the method of fundamental solutions (see the general 2D case here) for the polygonal case. The method of fundamental solutions consists in finding a function which already satisfies the equation on the whole plane, and see that it is zero on the boundary of the desired shape. We choose the fundamental solutions as being the radial functions where are some well chosen source points and . We search our solution as a linear combination of the functions , so we will have to solve a system of the form

in order to find the desired eigenfunction. Since we cannot solve numerically this system for every we choose a discretization on the boundary of and we arrive at a system of equations like:

and this system has a nontrivial solution if and only if the matrix is singular. The values for which is singular are exactly the square roots of the eigenvalues of our domain .

## Cyclic polygon has largest area

Suppose are positive real numbers. Then the following statements are equivalent:

1. satisfy the inequalities

2. There exists a polygon with sides .

3. There exists a convex polygon with sides .

4. There exists a cyclic polygon with sides .

Of all these, the cyclic polygon has the maximal area.

## Isoperimetric inequality for polygons

Among polygons with sides with given perimeter , the one which maximizes the area is regular.

## Regular polygons and lattice points.

Se say that a point in the plane is a lattice point if both its coordinates are integers. Prove that for we can’t find a regular polygon with egdes, its vertices being lattice points.

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