Triangular and quadrangular pyramids in a three-dimensional normed space

The main results are as follows.
Every three-dimensional real normed space contains an isometrically embedded set of vertices of a Euclidean tetrahedron whenever the ratio of lengths for each pair of edges of the tetrahedron is $\ge(\sqrt{8/3}+1)/3<0.878$.
Every three-dimensional normed space contains an affine image of a regular quadrangular pyramid having lateral edges of equal length, base edges of equal length, and base diagonals of equal length, having a predetermined ratio $>\sqrt{2/3}$ of the length of the lateral edge to the length of the base edge.