On the affine convexity of convex curves and hypersurfaces (Q1610969)

The author considers interesting analogies between Euclidean and equi-affine convex geometry. In analogy to the ordinary convexity of ordinary polygons in \(\mathbb E^2\), a closed polygon in \(\mathbb E^2\) consisting of finitely many parabolic arcs is affinely convex if each parabola containing one of the parabolic arcs supports the polygon. As a limiting case, a smooth oval in \(\mathbb E^2\) is affinely convex, if each hyperosculating parabola supports the oval. Theorem: An oval of class \({\mathcal C}^4\) is affinely convex if and only if its affine curvature is nonnegative. Unfortunately the second definition cannot be extended to higher dimensions, but one can define an ovaloid in \(\mathbb E^d\) of class \({\mathcal C}^5\) to be affinely convex, if all its affine principal curvatures are non-negative. Finally, the affine surface area of ovaloids in \(\mathbb E^d\) of class \({\mathcal C}^5\) is defined as usual. Theorem: For ovaloids in \(\mathbb E^d\) of class \({\mathcal C}^5\) which are affinely convex, the affine surface area is monotone (with respect to inclusions) and continuous (with respect to the Hausdorff metric). Compare also the author's book [`Affine geometry of convex bodies' (1998; Zbl 0899.52005)].