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The geometry of triangles and convex optimization get on well together

Nothing in the world happens without optimization, and there is no doubt that all aspects of the world that have a rational basis can be explained by optimization methods" used to say L.Euler in 1744. This is con_rmed by numerous applications of mathematics in physics, mechanics, economy, etc. This also holds true for the classical plane geometry, especially the one dealing with triangles. In this work, we show how (classical) geometry of triangles and (modern) convex optimization get on well together. In what constitutes the essential of our approach, we revisit the particular points of a triangle (also called triangle centers), at least the best known ones, in the sole light of contiuous optimization. We put into perspective how these points are the unique solutions of appropriate minimization problems, where objective functions are built up from distances to sides of the triangle or to vertices of the triangle. These objective functions all are convex, even strictly convex, but, to encompass all the encountered situations, one should however make a concession: to accept nonsmooth convex functions. Hence, we get at variational characterizations of the main triangle centers; among them, the orthocenter turned out to be the toughest to tackle . In doing so, we also have added a new triangle center to an existing long series  via its variational formulation. Key-words. Particular points in a triangle. Euclidean distance between two points, from a point to a line. Convex function of several variables. Gradient of a function. Subdi_erential of a convex function. Optimality conditions.

Jean-Baptiste Hiriart-Urruty, 法国图卢兹Paul Sabatier大学数学研究所教授、法国著名运筹学专家及数学教育家。在泛函分析杂志等一些国际著名数学杂志上发表学术论文一百余篇、多部数学著作被国际学术界广泛引用。