### G-2012-22

# The Small Octagons of Maximal Width

## Charles Audet, Pierre Hansen, Frédéric Messine, and Jordan Ninin

The paper answers an open problem introduced by Bezdek and Fodor in 2000. The width of any unit-diameter octagon is shown to be less than or equal to `\(\frac{1}{4}\sqrt{10 + 2\sqrt7}\)`

and there are infinitely many small octagons having this optimal width. The proof combines geometric and analytical reasoning as well as the use of a recent version of the deterministic and reliable global optimization code IBBA based on interval and affine arithmetics. The code guarantees a certified numerical accuracy of `\(1\times10^{-7}\)`

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Published **May 2012**
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14 pages