Eutactic star

A star is eutactic if it is the orthogonal projection of plus and minus the set of standard basis vectors (i.e., the vertices of a cross-polytope) from a higher-dimensional space onto a subspace.

Such stars were called "eutactic" – meaning "well-situated" or "well-arranged" – by Schläfli (1901, p. 134) because, for a common scalar multiple, their vectors are projections of an orthonormal basis.

Hadwiger's principal theorem states that the vectors ±a1, ..., ±as form a eutactic star if and only if there is a constant ζ such that Tx = ζx for every x.

Hadwiger's theorem implies the equivalence of Schläfli's stipulation and the geometrical definition of a eutactic star, by the polarization identity.

Furthermore, both Schläfli's identity and Hadwiger's theorem give the same value of the constant ζ. Eutactic stars are useful largely because of their relationship with the geometry of polytopes and groups of orthogonal transformations.