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convex function; convex set; exceptional set
We construct a Lipschitz function on $\mathbb R^2$ which is locally convex on the complement of some totally disconnected compact set but not convex. Existence of such function disproves a theorem that appeared in a paper by L. Pasqualini and was also cited by other authors.
[1] Burago Ju D., Zalgaller V.A.: Sufficient tests for convexity. Zap. Naucn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. 45 (1974), 3–52. MR 0377693
[2] Dmitriev V.G.: On the construction of $\mathcal H_{n-1}$-almost everywhere convex hypersurface in $\mathbb R^{n+1}$. Mat. Sb. (N.S.) 114(156) (1981), 511–522. MR 0615339
[3] Kirszbraun M.D.: Über die zusammenziehende und Lipschitzsche Transformationen. Fund. Math. 22 (1934), 77–108.
[4] Pasqualini L.: Sur les conditions de convexité d'une variété. Ann. Fac. Sci. Toulouse Sci. Math. Sci. Phys. (4) 2 (1938), 1–45. MR 1508453 | Zbl 0026.08801
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