Moscow--Petersburg seminar on low-dimensional mathematics SPb, PDMI, 11.10.2002, 16:00--17:45, room 311 "Topology of corank 1 singularities of a generic wave front" V. D. Sedykh (Moscow) New conditions of the coexistence of stable corank 1 Legendre singularities are found. These conditions lead to nontrivial corollaries in analysis and geometry. For example, let a smooth closed connected generic curve $\gamma$ in ${\Bbb R}^9$ be Barner convex (for any $8$ of its points, taking the multiplicities into account, there exists a hyperplane which intersects $\gamma$ only at these points). Denote by $\chi(A_{\mu_1}+\dots+A_{\mu_m})$ the number of supporting hyperplanes which are tangent to $\gamma$ at $m$ points with multiplicities $\mu_1,\dots,\mu_m$ such that $\mu_1+\dots+\mu_m=9$. Then $$ \begin{array}{r} 14\chi(A_9)-5\chi(A_7+2A_1)-2\chi(A_5+A_3+A_1)+2\chi(A_5+4A_1)- {}\hspace{0.65cm}\\ {}-\chi(3A_3)+\chi(2A_3+3A_1)- \chi(A_3+6A_1)+\chi(9A_1)=140\\ \end{array} $$ where $kA_{\mu}=A_{\mu}+\dots+A_{\mu}$ ($k$ times).