Geodesic
Equilateral simplices in normed 4-space
Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 9, Tome 329 (2005), pp. 88-91

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Let $E$ be a 4-dimensional real normed space, $x\ge3/4$ a positive number, and $P\subset E$ a 3-plane. It is proved that there exist 4 equidistant points $A_1$, $A_2$, $A_3$, $A_4\in P$ and a point $A_5\in E$ such that $\|A_5A_i\|=x\cdot\|A_1A_2\|$ for $i=1,2,3,4$. In particular, $E$ contains an equilateral simplex. 
@article{ZNSL_2005_329_a7,
     author = {V. V. Makeev},
     title = {Equilateral simplices in normed 4-space},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {88--91},
     publisher = {mathdoc},
     volume = {329},
     year = {2005},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2005_329_a7/}
}
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V. V. Makeev. Equilateral simplices in normed 4-space. Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 9, Tome 329 (2005), pp. 88-91. http://geodesic.mathdoc.fr/item/ZNSL_2005_329_a7/