Welcome to MathOverflow

Question

0

Let me state my problem. Suppose we have a ball $B$ in standard $\mathbb{R}^3$, that is a $\varepsilon$-neighbourhood of $0$ point. Suppose we have a family of cones $X_C = \lbrace C > 0 \vert x^2 + y^2 \leqslant C \cdot z^2 \rbrace $. Also we have a homeomorphism $h$ that maps $B$ on itself ($h(B) = B$) and $h(0) = 0$ (if it's crucial, $h(B \cap Oz) = B \cap Oz$ and $h(0) = 0$). So, the question is: if we take a cone, corresponding to some value $C_1$, does exist some $C_2$ that $f(X_{C_1}) \subseteq X_{C_2}$?

edited Jul 11 at 19:42

asked Jul 11 at 16:21

Evgeny
3●2

	Does $Oz$ mean the $z$-axis intersected with the ball $B$? – Lee Mosher Jul 11 at 16:35
	Whether it does or not, though, this questions seems to me to be more suited to other sites such as math.stackexchange.com. It is not a research level question, which is what Math Overflow is for, as you will see if you read the faq. – Lee Mosher Jul 11 at 16:59
	Yes, I meant that. Sorry, I'll correct this moment – Evgeny Jul 11 at 17:00
	That question appeared in my research, so I just wanted to check is this a common fact (that I've missed in my education) or should I cope with that question myself. – Evgeny Jul 11 at 17:05
	I guess you meant $h(X_{C^}) \subseteq X_{C^}$! – McParson Jul 11 at 17:31

show 3 more comments

Lee Mosher · Accepted Answer · 2012-07-11 20:11:58Z UTC

0

Here is the counterexample, re-expressed. Let $S_r$ be the sphere of radius $r \epsilon$. Construct a homeomorphism $h : B \to B$ with the following properties:

$h(S_r) = S_r$ for each $r \in [0,1]$
$h(S_{1/3} \cap X_{C^*}) = S_{1/3} \cap X_{C^*}$
$h(S_{2/3} \cap X_{C^*}) \supset S_{2/3} \cap (\mathbb{R}^3 - X_{C^*})$
And, if you like, $h$ preserves the $z$-axis, which prevents the last $\supset$ from being $=$.

answered Jul 11 at 20:11

Lee Mosher
5,806●1●9●26

Well, thanks for idea, I've constructed one example of that kind and my question has negative answer there; but I couldn't preserve $z$-axis. So far, I've discovered another example by myself that preserves $z$-axis, but too has negative answer. I just took upper half of cone and laid it on disc $B \cap Oxy$ (and constructed images for other parts of ball correspondingly). So, thank you for idea that my question doesn't have positive answer in all cases! – Evgeny Jul 11 at 20:30

Does homeomorphism preserves the family of cones?

Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

1 Answer

You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.