arXiv subjects

Chiara Meroni

Publications and source records attributed to Chiara Meroni.

22 records · Page 2Linked to original sources

arXiv2022

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.

math.AG↗

arXiv2022

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 $3$, we give a characterization of manifolds having a Heegaard splitting of genus $g$ in terms of the existence of certain Reeb functions. Similar results are proved in dimension $n\geq 5$.

math.GT↗

arXiv2022

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.

math.AG↗

arXiv2021

Real Lines on Random Cubic Surfaces

We give an explicit formula for the expectation of the number of real lines on a random invariant cubic surface, i.e. a surface $Z\subset \mathbb{R}P^3$ defined by a random gaussian polynomial whose probability distribution is invariant under the action of the orthogonal group $O(4)$ by change of variables. Such invariant distributions are completely described by one parameter $λ\in [0,1]$ and as a function of this parameter the expected number of real lines equals: \begin{equation} E_λ=\frac{9(8λ^2+(1-λ)^2)}{2λ^2+(1-λ)^2}\left(\frac{2λ^2}{8λ^2+(1-λ)^2}-\frac{1}{3}+\frac{2}{3}\sqrt{\frac{8λ^2+(1-λ)^2}{20λ^2+(1-λ)^2}}\right). \end{equation} This result generalizes previous results by Basu, Lerario, Lundberg and Peterson for the case of a Kostlan polynomial, which corresponds to $λ=\frac{1}{3}$ and for which $E_{\frac{1}{3}}=6\sqrt{2}-3.$ Moreover, we show that the expectation of the number of real lines is maximized by random purely harmonic cubic polynomials, which corresponds to the case $λ=1$ and for which $E_1=24\sqrt{\frac{2}{5}}-3$.

math.AG↗