## M. Nevskii and A. Ukhalov, "Perfect simplices in \$\mathbb{R}^5\$", Beitr. Algebra Geom., 59:3, 501--521 (2018).

M. Nevskii and A. Ukhalov, "Perfect simplices in \$\mathbb{R}^5\$", Beitr. Algebra Geom., 59:3, 501--521 (2018).

АННОТАЦИЯ

Let Qn=[0,1]nQn=[0,1]n be the unit cube in RnRn, n∈Nn∈N. For a nondegenerate simplex S⊂RnS⊂Rn, consider the value ξ(S)=min{σ>0:Qn⊂σS}ξ(S)=min{σ>0:Qn⊂σS}. Here σSσS is a homothetic image of S with homothety center at the center of gravity of S and coefficient of homothety σσ. Let us introduce the value ξn=min{ξ(S):S⊂Qn}ξn=min{ξ(S):S⊂Qn}. We call S a perfect simplex if S⊂QnS⊂Qn and QnQn is inscribed into the simplex ξnSξnS. It is known that such simplices exist for n=1n=1 and n=3n=3. The exact values of ξnξnare known for n=2n=2 and in the case when there exists an Hadamard matrix of order n+1n+1; in the latter situation ξn=nξn=n. In this paper we show that ξ5=5ξ5=5 and ξ9=9ξ9=9. We also describe infinite families of simplices S⊂QnS⊂Qn such that ξ(S)=ξnξ(S)=ξn for n=5,7,9n=5,7,9. The main result of the paper is the confirmation of the existence of perfect simplices in R5R5.

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