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Projective convexity in P³ implies Grassmann convexity

By B. Shapiro and M. Shapiro


In this note we introduce the notion of Grassmann convexity analogous to the wellknown notion of convexity for curves in real projective spaces. We show that the curve in G2,4 osculating to a convex closed curve in P³ is Grassmann-convex. This proves that the tangent developable (i.e. the hypersurface formed by all tangents) of any convex curve in P³ has the ‘degree’ equal to 4. Here by ‘degree’ of a real projective hypersurface we mean the maximal total multiplicity of its intersection with a line

Year: 2011
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