32 research outputs found
On the convex hull of a space curve
The boundary of the convex hull of a compact algebraic curve in real 3-space
defines a real algebraic surface. For general curves, that boundary surface is
reducible, consisting of tritangent planes and a scroll of stationary
bisecants. We express the degree of this surface in terms of the degree, genus
and singularities of the curve. We present algorithms for computing their
defining polynomials, and we exhibit a wide range of examples.Comment: 19 pages, 4 figures, minor change
Semidefinite representation of convex hulls of rational varieties
Using elementary duality properties of positive semidefinite moment matrices
and polynomial sum-of-squares decompositions, we prove that the convex hull of
rationally parameterized algebraic varieties is semidefinite representable
(that is, it can be represented as a projection of an affine section of the
cone of positive semidefinite matrices) in the case of (a) curves; (b)
hypersurfaces parameterized by quadratics; and (c) hypersurfaces parameterized
by bivariate quartics; all in an ambient space of arbitrary dimension
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
Convex hulls of curves of genus one
Let C be a real nonsingular affine curve of genus one, embedded in affine
n-space, whose set of real points is compact. For any polynomial f which is
nonnegative on C(R), we prove that there exist polynomials f_i with f \equiv
\sum_i f_i^2 (modulo I_C) and such that the degrees deg(f_i) are bounded in
terms of deg(f) only. Using Lasserre's relaxation method, we deduce an explicit
representation of the convex hull of C(R) in R^n by a lifted linear matrix
inequality. This is the first instance in the literature where such a
representation is given for the convex hull of a nonrational variety. The same
works for convex hulls of (singular) curves whose normalization is C. We then
make a detailed study of the associated degree bounds. These bounds are
directly related to size and dimension of the projected matrix pencils. In
particular, we prove that these bounds tend to infinity when the curve C
degenerates suitably into a singular curve, and we provide explicit lower
bounds as well.Comment: 1 figur
Polytopes from Subgraph Statistics
Polytopes from subgraph statistics are important in applications and
conjectures and theorems in extremal graph theory can be stated as properties
of them. We have studied them with a view towards applications by inscribing
large explicit polytopes and semi-algebraic sets when the facet descriptions
are intractable. The semi-algebraic sets called curvy zonotopes are introduced
and studied using graph limits. From both volume calculations and algebraic
descriptions we find several interesting conjectures.Comment: Full article, 21 pages, 8 figures. Minor expository update
Algebraic Boundaries of Convex Semi-algebraic Sets
We study the algebraic boundary of a convex semi-algebraic set via duality in
convex and algebraic geometry. We generalize the correspondence of facets of a
polytope to the vertices of the dual polytope to general semi-algebraic convex
bodies. In the general setup, exceptional families of extreme points might
exist and we characterize them semi-algebraically. We also give an algorithm to
compute a complete list of exceptional families, given the algebraic boundary
of the dual convex set.Comment: 13 pages, 2 figures; Comments welcom
Properties of a curve whose convex hull covers a given convex body
In this note, we prove the following inequality for the norm of a convex body
in , : , where is the diameter of ,
is any curve in whose convex hull covers , and
is the gamma function. If in addition has constant width ,
then we get the inequality . In addition, we pose several unsolved
problems.Comment: 7 page