Thickenings of a metric space capture local geometric properties of the space. Here we exhibit applications of lower bounding the topology of thickenings of the circle and more generally the sphere. We explain interconnections with the geometry of circle actions on Euclidean space, the structure of zeros of trigonometric polynomials, and theorems of Borsuk-Ulam type. We use the combinatorial and geometric structure of the convex hull of orbits of circle actions on Euclidean space to give geometric proofs of the homotopy type of metric thickenings of the circle. Homotopical connectivity bounds of thickenings of the sphere allow us to prove that a weighted average of function values of odd maps @math on a small diameter set is zero. We prove an additional generalization of the Borsuk-Ulam theorem for odd maps @math . We prove such results for odd maps from the circle to any Euclidean space with optimal quantitative bounds. This in turn implies that any raked homogeneous trigonometric polynomial has a zero on a subset of the circle of a specific diameter; these results are optimal.
@cite_38: We consider the convex hull B_k of the symmetric moment curve U(t)=(cos t, sin t, cos 3t, sin 3t, ..., cos (2k-1)t, sin (2k-1)t) in R^ 2k , where t ranges over the unit circle S= R 2pi Z. The curve U(t) is locally neighborly: as long as t_1, ..., t_k lie in an open arc of S of a certain length phi_k>0, the convex hull of the points U(t_1), ..., U(t_k) is a face of B_k. We characterize the maximum possible length phi_k, proving, in particular, that phi_k > pi 2 for all k and that the limit of phi_k is pi 2 as k grows. This allows us to construct centrally symmetric polytopes with a record number of faces.
@cite_13: Let @math XźA2r be a real curve embedded into an even-dimensional affine space. We characterise when the @math rth secant variety to @math X is an irreducible component of the algebraic boundary of the convex hull of the real points @math X(R) of @math X. This fact is then applied to @math 4-dimensional @math SO(2)-orbitopes and to the so called Barvinok---Novik orbitopes to study when they are basic closed semi-algebraic sets. In the case of @math 4-dimensional @math SO(2)-orbitopes, we find all irreducible components of their algebraic boundary.
@cite_11: Here we study the kth symmetric trigonometric moment curve and its convex hull, the Barvinok–Novik orbitope. In 2008, Barvinok and Novik introduced these objects and showed that there is some threshold so that for two points on @math with arclength below this threshold the line segment between their lifts to the curve forms an edge on the Barvinok–Novik orbitope, and for points with arclength above this threshold their lifts do not form an edge. They also gave a lower bound for this threshold and conjectured that this bound is tight. Results of Smilansky prove tightness for k=2. Here we prove this conjecture for all k.
Though the facial structure of the Barvinok--Novik orbitopes @math is not known for @math , certain neighborliness results have been established @cite_38 . Sinn has shown that the orbitopes are simplicial @cite_13 . Additionally, Vinzant proved that the edges of @math consist of all line segments @math with @math @cite_11 . In other words, the edges of @math are the same as the edges of @math . The following is an immediate corollary of the work of Sinn and Vinzant.