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Topic: Clarification on the result of the equichordal point problem
Replies: 6   Last Post: Jan 5, 2004 3:09 PM

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 John Conway Posts: 2,238 Registered: 12/3/04
Re: Clarification on the result of the equichordal point problem
Posted: Dec 31, 2003 7:51 PM

On 31 Dec 2003, Shinya wrote:

> Hello. While I was doing research on the equichordal point problem,
> I encountered a confusion.

I might be able to clear it up, since I've been interested in this
problem for more than 40 years, and know the state of play. I'll read
on...

> Equichordal Point Problem: Can a plane region have two equichordal
> points?

The answer is definitely "No", as was first established by Rychlik

> The source of the confusion is Mathworld. On the one hand, it states
> that the problem was fully solved by Rychlik in 1997

So that's the right one, while in this one:

> on the other hand, it states that "a plane convex region can
> have two equichordal points."

the word "can" is obviously a typographical error for "cannot".

This particular error ("can" for "cannot") is really quite
common. There have been 5 or 6 different instances that have
puzzled me for quite considerable times!

> By the way, the problem was solved in the negative: This problem has
> been fully solved in the negative by the author of this announcement
> just recently. (From abstract of the original paper) i.e.,
>
> No Jordan curve may contain two or more equichordal points.
>
> Could anyone help me?

You seem to have helped yourself pretty well, but might like
to have the reassurance that your guess is the right one.

Regards, John Conway

Date Subject Author
12/31/03 Hisanobu Shinya
12/31/03 John Conway
1/1/04 Paulo Santa Rita
1/5/04 Michael Lambrou
1/5/04 erdosfan
1/5/04 Michael Lambrou
1/5/04 erdosfan