Criterion for $$k$$-Regularity of a Mapping in Terms of a Space of Measures

The question of the existence of a $$k$$ -regular embedding of a given finite-dimensional compact metric space $$X$$ into a Euclidean space of a given dimension $$m$$ is studied, i.e., such a continuous mapping that no more than $$k$$ points fall on any plane of dimension $$(k-1)$$ . These mappings are important in the theory of interpolation and approximation.