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 $1\times10^{-7}$.