Using Lie groupoids, we prove that the injectivity radius of a manifold with a Lie structure at infinity is positive. This relies on the integrability of the corresponding Lie algebroid, a well-known result that we prove explicitly by regarding manifolds with corners as particular instances of orbifolds. Le rayon d'injectivité des variétés munies d'une structure de Lie à l'infiniRésumé À l'aide des groupoïdes de Lie, on montre que le rayon d'injectivité d'une variété munie d'une structure de Lie à l'infini est strictement positif. La démonstration s'appuie sur l'intégrabilité de l'algébroïde de Lie correspondant, un résultat bien connu que l'on établit directement en regardant les variétés à coins comme des cas particuliers d'orbifolds.
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