Re: Roundess as a shape coefficent

On 27 jul, 12:33, "r.p.mac...@xxxxxxxxxxxxxx"
<r.p.mac...@xxxxxxxxxxxxxx> wrote:
Roundness of the object bounded with closed curve is defined as:
r = perimeter^2/(4*\pi*area)
and is said to be minimal and equals 1 for a perfect circle.

Do you know (or at least where to find) the proof that this is indeed
minimal?

This would be the solution for task of finding maximal area that we
can bound closing the string of given length and for some integral
inequities as well (to which it leads when we substitute formulas for
perimeter and area).

Thank for all responses

http://en.wikipedia.org/wiki/Isoperimetric_inequality

From:

http://www-groups.dcs.st-and.ac.uk/~history/Extras/Calculus_of_Variations.html

Quote:

for today we know: Of all surfaces bounded by curves of a given
length,
the circle is the one of largest area. The branch of mathematics which
establishes a rigorous proof of this statement is the calculus of
variations.

http://en.wikipedia.org/wiki/Calculus_of_variations

Could'nt find back an old reproduction of the relevant proof. Sorry.

Han de Bruijn
.

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