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Following Fr\"ohlich and Spencer, we study one dimensional Ising spin systemswith ferromagnetic, long range interactions which decay as $|x-y|^{-2+\alpha}$,$0\leq \alpha\leq 1/2$. We introduce a geometric description of the spinconfigurations in terms of triangles which play the role of contours and forwhich we establish Peierls bounds. This in particular yields a direct proof ofthe well known result by Dyson about phase transitions at low temperatures.Comment: 28 pages, 3 figure

Geometry of contours and Peierls estimates in d=1 Ising models with long range interactions