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The convexity of a discrete region is a property used in numerous domains of computational imagery. We study its detection in the particular case of polyominoes. We present a first method, directly relying on its definition. A second method, which is based on techniques for segmentation of curves in discrete lines, leads to a very simple, linear, algorithm whose correctness is proven. Correlatively, we obtain a characterisation of lower and upper frontiers of the convex hull of a discrete line segment. Finally, we evoke some applications of these results to the problem of discrete tomography. [Copyright &y& Elsevier]