Many new universal relations are obtained between the Euler numbers of manifolds of singular supporting hyperplanes of an arbitrary generic smooth closed $k$-dimensional submanifold in ${{\mathbb R}}^n$ where $n\leq 7$ or $k=1$. These relations are applied to Barner-convex curves in an odd-dimensional space ${{\mathbb R}}^n$. A universal (nontrivial) linear relation is established between the numbers of singular supporting hyperplanes of various types but of the same total multiplicity $n$ of tangency with a given generic smooth closed connected Barner-convex curve in ${{\mathbb R}}^n$. The coefficients of this relation are defined by Catalan numbers.