## 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.

## Agreg 2012 Analysis Part 4

**A Fixed Point Theorem**

This parts wishes to extend the next result (which can be used without proof) to infinite dimension.

**Theorem.** (Browuer) Consider a finite dimensional normed vector space. Consider a convex, closed, bounded non-void set. If is a continuous application, then has a fixed point in .

1. In endowed with we consider the following application:

Prove that is continuous with values in the unit sphere of , but does not admit any fixed points.

2. Consider a normed vector space, a closed, bounded non-void subset of and a compact application (not necessarily linear). (a compact application maps bounded sets into relatively compact sets)

i) Let . We can cover (which is compact) by a finite number of open balls of radius : with for every . For we define

Prove that is continuous and that there exists such that for we have .

ii) We introduce the application defined by

Prove that for every we have .

## Darboux Functions with no Iterate Fixed Points

It is well known that there exist functions which have Darboux property and they have no fixed points. An example can be found in an earlier post of mine. Here is a generalization of that result.

There exist functions which have Darboux property and for which none of its iterates has a fixed point, i.e. for every and for every .

## Brouwer fixed point theorem

Let be an open subset of such that is homeomorphic to the closed unit ball ( in ). If (continuous function on ) and , then has a fixed point in .

## The Existence of a Triangle with Prescribed Angle Bisector Lengths

Prove that for any there exists a unique triangle (up to an isometry) such that are the lengths of bisectors of this triangle

*Solved by L. Panaitopol & P. Mironescu in 1994, AMM 101, 58-60*

Read more…

## Fixed point for an operator

Suppose is a Hilbert space and is a linear operator on with . Prove that if and only if .

**Solution:** Using the Cauchy-Buniakovski inequality, we get

. Since this implies equality in the C-B inequality, we must have . We easily find that . The converse is equivalent to the implication above.

## Fixed point theorem

Suppose is a compact convex set and is a function satisfying . Prove that has a fixed point ( i.e. with ). Read more…