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 R~n where n ≤ 7 or k = 1. These relations are applied to Barner-convex curves in an odd-dimensional space 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 R~n. The coefficients of this relation are defined by Catalan numbers.
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