Triangular and Quadrangular Pyramids in a Three-Dimensional Normed Space

Abstract

The main results are as follows. Let T be a Euclidean tetrahedron such that the ratio of lengths in each pair of edges of T is at least \( \left(\sqrt{8/3}+1\right)/3<0.878 \). Then each three-dimensional real normed space contains an isometrically embedded set of vertices of T . Let E be a three-dimensional normed space, and let x be a preassigned real number greater than \( \sqrt{2/3} \). Then E contains an affine image Π of a regular quadrangular pyramid such that the lateral edges of Π have equal length, the base edges of Π have equal length, the base diagonals also have equal length, and the ratio between the length of the lateral edges and the length of the base edges is equal to x. Bibliography: 5 titles.