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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
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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

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

The answer is definitely "No", as was first established by Rychlik
about 5 years ago. I read on again...

> 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

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