Geodesic
On simplexes inscribed in a~hypersurface
Matematičeskie zametki, Tome 5 (1969) no. 1, pp. 81-89

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The sufficient conditions are obtained for the existence, on a hyper surface $M\subset R^n$, of $k$ points whose convex hull forms a $(k-1)$-dimensional simplex, homothetic to a given simplex $\Delta\subset R^n$. In particular, it is shown that if $M$ is a smooth hypersurface, homeomorphic to a sphere, such points will exist for any simplex $\Delta\subset R^n$. The proofs are based on simple topological considerations. There are six references. 
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     author = {M. L. Gromov},
     title = {On simplexes inscribed in a~hypersurface},
     journal = {Matemati\v{c}eskie zametki},
     pages = {81--89},
     publisher = {mathdoc},
     volume = {5},
     number = {1},
     year = {1969},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1969_5_1_a10/}
}
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M. L. Gromov. On simplexes inscribed in a~hypersurface. Matematičeskie zametki, Tome 5 (1969) no. 1, pp. 81-89. http://geodesic.mathdoc.fr/item/MZM_1969_5_1_a10/