The topological type of the α-sections of convex sets

Envelopes of $$\alpha $$ α -Sections

“…There is a vast literature on these subjects; we refer in the following to very few articles, and briefly present even fewer, that we find particularly relevant for our study. Further references can be found in those papers.Generalizing previous results on common tangents and common transversals to families of convex bodies [2], [6], J. Kincses [17] showed that, for any well-separated family of strictly convex bodies, the space of α-sections is diffeomorphic to S d−k .A billiard table is a planar strictly convex body K. Choose a starting point x outside the table and one of the two tangents through x to K, say the right one, denoted by D; the image T (x) of x by the billiard map T is the point symmetric of x with respect to the tangency point D ∩ ∂K. A caustic of the billiard is an invariant curve (an invariant torus in the terminology of the KAM theory).…”

“…Generalizing previous results on common tangents and common transversals to families of convex bodies [2], [6], J. Kincses [17] showed that, for any well-separated family of strictly convex bodies, the space of α-sections is diffeomorphic to S d−k .…”

Spherical floating bodies

“…1 dx on the unit sphere. Directional derivative of this function was already calculated in [12] and later in [9] but with different notations and purpose, so we decided to present the following short calculation here. Fix an arbitrary unit vector u and choose arbitrarily an other unit vector u ⊥ orthogonal to u.…”

Common tangents to polytopes

“…In any case, there are now different proofs and also several generalizations of this theorem, see e.g. [1], [8], [9], and [10].…”