Threadable Curves

We define a plane curve to be threadable if it can rigidly pass through a point-hole in a line L without otherwise touching L. Threadable curves are in a sense generalizations of monotone curves. We have two main results. The first is a linear-time algorithm for deciding whether a polygonal curve is threadable---O(n) for a curve of n vertices---and if threadable, finding a sequence of rigid motions to thread it through a hole. We also sketch an argument that shows that the threadability of algebraic curves can be decided in time polynomial in the degree of the curve. The second main result is an O(n polylog n)-time algorithm for deciding whether a 3D polygonal curve can thread through hole in a plane in R^3, and if so, providing a description of the rigid motions that achieve the threading.Comment: 16 pages, 12 figures, 12 references. v2: Revised with brief addendum after Mikkel Abrahamsen pointed us to a relevant reference on "sweepable polygons." v3: Major revisio

Orbitopes

An orbitope is the convex hull of an orbit of a compact group acting linearly on a vector space. These highly symmetric convex bodies lie at the crossroads of several fields, in particular convex geometry, optimization, and algebraic geometry. We present a self-contained theory of orbitopes, with particular emphasis on instances arising from the groups SO(n) and O(n). These include Schur-Horn orbitopes, tautological orbitopes, Caratheodory orbitopes, Veronese orbitopes and Grassmann orbitopes. We study their face lattices, their algebraic boundary hypersurfaces, and representations as spectrahedra or projected spectrahedra.Comment: 37 pages. minor revisions of origina

The Convex Hull of a Variety

We present a characterization, in terms of projective biduality, for the hypersurfaces appearing in the boundary of the convex hull of a compact real algebraic variety.Comment: 12 pages, 2 figure