Weak convexity does not imply convexity for curves in \(\mathbb{R} P^n,n>2\)
An osculating hyperplane at a point of a curve in \(\mathbb{R} P\) (in \(\mathbb{R}^n)\) is a hyperplane whose multiplicity of intersection with the curve at that point is at least \(n\). A flattening of a curve in \(\mathbb{R} P^n\) (in \(\mathbb{R}^n)\) is a point at which the multiplicity of intersection with its osculating hyperplane is at least \(n+1\), whereas at an ordinary point it is \(n\). A smooth curve in \(\mathbb{R} P^n\) is called convex if any hyperplane intersects it in at most \(n\) points, taking into acount multiplicities. A curve in \(\mathbb{R} P^2\) (in \(\mathbb{R}^2)\) is convex if and only if it has no flattening and its osculating hyperplane intersects it only at the point of osculation. In [Transl., Ser. 2, Am. Math. Soc. 171, 11-22 (1996; Zbl 0973.53502) and Russ. Math. Surv. 51, 1-47 (1996; Zbl 1053.37500)], \textit{V. I. Arnol'd} posed the problem to decide whether these two properties imply convexity for dimensions greater than 2. In the present paper the author proves that for \(n>2\), the answer to Arnol'd's question is negative.
convex curve in \(n\)-dimensional real projective space
scientific article; zbMATH DE number 1777363