Regions in $\mathbb{C}$ containing rectangles

Question

Can you find an open connected subset $\Omega$ in $\mathbb{C}$ such that it has the following property?

If $a,b,c \in \Omega$ are the vertices of a right triangle, then the rectangle given by $a,b,c$ lies in $\Omega$.

I suspect that the only such set is $\mathbb{C}$ itself, but I am unable to prove this, beyond some handwaving about taking points near the boundary.

Any suggestions?

I didnt know what to tag this as, so I put geometry. Please tag if you think something is more suitable. — , Feb 17 '14 at 07:09
It doesn't seem to be $\mathbb{C}$ only, for example any rectangle does have this property. — Marcin Łoś, Feb 17 '14 at 07:16
@Marcin Łoś No, if $\Omega$ is the rectangle $1,i,-1,-1$ an the right triangle is $0.9, 0, 0.9i$ the 4th vertex is $0.9+0.9i$ which is not in $\Omega$ — miracle173, Feb 17 '14 at 08:18
@5xum No, if $\Omega$ is the rectangle $1,i,-1,-1$ an the right triangle is $0.9, 0, 0.9i$ the 4th vertex is $0.9+0.9i$ which is not in $\Omega$ — miracle173, Feb 17 '14 at 08:18
@miracle173 Right, somehow I assumed the edges are parallel to the axes. — Marcin Łoś, Feb 17 '14 at 09:10

score 5 · Accepted Answer · answered Feb 17 '14 at 07:22

5

Assume $a\in\Omega\ne\mathbb C$. The complement is closed hence there exists $b\notin\Omega$ that minimizes $d(a,b)$. Consider the square with $ab$ as diagonal. Then the other two vertices are closer to $a$ than $b$, hence are in $\Omega$, hence this is a square with three vertices in $\Omega$, contradiction! (We see that general rectangles are not needed, squares suffice)

answered Feb 17 '14 at 07:22

Hagen von Eitzen

374,180

So it is not necessary for $\Omega$ to be connected? – miracle173 Feb 17 '14 at 07:44
1

@miracle173 In fact it is sufficient that the complement $\mathbb C-\Omega$ is not dense in $\mathbb C$ (i.e $\Omega$ has at least one interior point $a$. If $r:=\inf{,d(a,x)\mid x\notin \Omega,}>0$ pick $b\notin \Omega$ with $d(a,b)<\sqrt 2 r$ and proceed as above. – Hagen von Eitzen Feb 17 '14 at 18:41

Regions in $\mathbb{C}$ containing rectangles

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