The weighted Minkowski type existence theorem for C-pseudo-cones with given weighted surface area measure was proved by Schneider. In this paper, we prove the existence of a solution to the weighted L_p Minkowski problem for C-pseudo-cones in the case of p≥ 1 .
Hadwiger’s theorem is a Helly-type theorem involving common transversals to families of convex sets instead of common intersections. Subsequently, Pollack and Wenger identified a necessary and sufficient condition, called a consistent k-ordering, for the existence of a hyperplane transversal for sets in ℝ^d . We obtain a quantitative generalization of Hadwiger’s theorem in ℝ^2 , showing that compact convex sets in ℝ^2 with a quantitative version of consistent ordering have a transversal satisfying quantitative requirements. Our proof generalizes the methods in Wenger’s proof of Hadwiger’s theorem in ℝ^2 . We also prove colorful versions of our results.
We study the geometric structure of Poncelet n-gons from a projective point of view. In particular we present explicit constructions of Poncelet n-gons for n = 7,8 and derive corresponding algebraic characterisations in terms of bracket polynomials, as well as describing a construction to produce a Poncelet 2n-gon from a starting Poncelet n-gon.
