Geodesic
On a theorem of Helly
Matematičeskie zametki, Tome 58 (1995) no. 6, pp. 818-827 Cet article a éte moissonné depuis la source Math-Net.Ru

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We consider a group of problems related to the well-known Helly theorem on the intersections of convex bodies. We introduce convex subsets $K(f)$ of a compact convex set $K$ defined by the relation $$ K(f)=\operatorname{co}\biggl\{\frac N{N+1}x+\frac 1{N+1}f(x)\biggr\} \quad(x\in K\subset\mathbb R^N), $$ where $f\colon K\to K$ are continuous mappings, and prove that the intersection $\bigcap_{f\in F}K(f)$ is not empty; here $F$ is the set of all continuous mappings $f\colon K\to K$. 
@article{MZM_1995_58_6_a1,
     author = {N. A. Bobylev},
     title = {On~a~theorem of {Helly}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {818--827},
     year = {1995},
     volume = {58},
     number = {6},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1995_58_6_a1/}
}
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N. A. Bobylev. On a theorem of Helly. Matematičeskie zametki, Tome 58 (1995) no. 6, pp. 818-827. http://geodesic.mathdoc.fr/item/MZM_1995_58_6_a1/

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