274 research outputs found
The geometry of discotopes
We study a class of semialgebraic convex bodies called discotopes. These are instances of zonoids, objects of interest in real algebraic geometry and random geometry. We focus on the face structure and on the boundary hypersurface of discotopes, highlighting interesting birational properties which may be investigated using tools from algebraic geometry. When a discotope is the Minkowski sum of two-dimensional discs, the Zariski closure of its set of extreme points is an irreducible hypersurface. In this case, we provide an upper bound for the degree of the hypersurface, drawing connections to the theory of classical determinantal varieties
Intersection Bodies of Polytopes: Translations and Convexity
We continue the study of intersection bodies of polytopes, focusing on the
behavior of under translations of . We introduce an affine hyperplane
arrangement and show that the polynomials describing the boundary of
can be extended to polynomials in variables within each
region of the arrangement. In dimension , we give a full characterization of
those polygons such that their intersection body is convex. We give a partial
characterization for general dimensions
On smooth functions with two critical values
We prove that every smooth closed manifold admits a smooth real-valued
function with only two critical values. We call a function of this type a
\emph{Reeb function}. We prove that for a Reeb function we can prescribe the
set of minima (or maxima), as soon as this set is a PL subcomplex of the
manifold. In analogy with Reeb's Sphere Theorem, we use such functions to study
the topology of the underlying manifold. In dimension , we give a
characterization of manifolds having a Heegaard splitting of genus in terms
of the existence of certain Reeb functions. Similar results are proved in
dimension
Convex Hulls of Curves: Volumes and Signatures
Taking the convex hull of a curve is a natural construction in computational
geometry. On the other hand, path signatures, central in stochastic analysis,
capture geometric properties of curves, although their exact interpretation for
levels larger than two is not well understood. In this paper, we study the use
of path signatures to compute the volume of the convex hull of a curve. We
present sufficient conditions for a curve so that the volume of its convex hull
can be computed by such formulae. The canonical example is the classical moment
curve, and our class of curves, which we call cyclic, includes other known
classes such as -order curves and curves with totally positive torsion. We
also conjecture a necessary and sufficient condition on curves for the
signature volume formula to hold. Finally, we give a concrete geometric
interpretation of the volume formula in terms of lengths and signed areas.Comment: 15 pages, 5 figures. Comments are welcome
Intersection Bodies of Polytopes
We investigate the intersection body of a convex polytope using tools from
combinatorics and real algebraic geometry. In particular, we show that the
intersection body of a polytope is always a semialgebraic set and provide an
algorithm for its computation. Moreover, we compute the irreducible components
of the algebraic boundary and provide an upper bound for the degree of these
components.Comment: As published in Beitr\"age zur Algebra und Geometrie. Correction of a
scaling factor from previous versio
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