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We establish near-optimal mixed-norm estimates for the X-ray transform restricted to polynomial curves with a weight that is a power of the affine arclength. The bounds that we establish depend only on the spatial dimension and the degree…
Quantitative bounds for random embeddings of $\mathbb{R}^{k}$ into Lorentz sequence spaces are given, with improved dependence on $\varepsilon$.
Systems describing the long-range interaction between individuals have attracted a lot of attention in the last years, in particular in relation with living systems. These systems are quadratic, written under the form of transport equations…
We prove explicit bounds on the number of lattice points on or near a convex curve in terms of geometric invariants such as length, curvature, and affine arclength. In several of our results we obtain the best possible constants. Our…
We prove tight upper bounds for the number of vertices of a simple polygon that is the union or the intersection of two simple polygons with given numbers of convex and concave vertices. The similar question on graphs of the lower (or…
We prove several interpolation results for holomorphic Legendrian curves lying in an odd dimensional complex Euclidean space with the standard contact structure. In particular, we show that an arbitrary countable set of points in…
Envelopes of parameterized families of plane curves is an important topic, both for the mathematics involved and for its applications. Nowadays, it is generally studied in a technology-rich environment, and automated methods are developed…
We explore the applications of Lorentzian polynomials to the fields of algebraic geometry, analytic geometry and convex geometry. In particular, we establish a series of intersection theoretic inequalities, which we call rKT property, with…
We give a combinatorial description of closed curves on oriented surfaces in terms of certain permutations, called charts. We describe automorphisms of curves in terms of charts and compute the total number of curves counted with…
We prove a number of results to the general effect that, under obviously necessary numerical and determinant constraints, "most" morphisms between fixed bundles on a complex elliptic curve produce (co)kernels which can either be specified…
We bound the number of incidences between points and spheres in finite vector spaces by bounding the sum of the number of points in the pairwise intersections of the spheres. We obtain new incidence bounds that are interesting when the…
For polyhedral convex cones in ${\mathbb R}^d$, we give a proof for the conic kinematic formula for conic curvature measures, which avoids the use of characterization theorems. For the random cones defined as typical cones of an isotropic…
The aim of this paper is to prove a uniform Fourier restriction estimate for certain $2-$dimensional surfaces in $\mathbb R^{2n}$. These surfaces are the image of complex polynomial curves $\gamma(z) = (p_1(z), \dots, p_n(z))$, equipped…
We study relations between maps between relatively hyperbolic groups/spaces and quasisymmetric embeddings between their boundaries. More specifically, we establish a correspondence between (not necessarily coarsely surjective)…
The Fr\'echet distance is a popular distance measure for curves which naturally lends itself to fundamental computational tasks, such as clustering, nearest-neighbor searching, and spherical range searching in the corresponding metric…
We consider two incidence problems for integral curves of vector fields. The first is an analogue of the Euclidean joints problem, in which lines are replaced by integral curves of smooth vector fields taken from some finite-dimensional…
In this paper, we continue the study of intersections of closed curves on translation surfaces, initiated in by S. Cheboui, A. Kessi and D. Massart for a family of arithmetic Veech surfaces and the author, E. Lanneau and D. Massart for a…
A self-avoiding plane-filling curve cannot be periodic, but we show that it can satisfy the local isomorphism property. We investigate three families of coverings of the plane by finite sets of nonoverlapping self-avoiding curves which…
For affine algebraic plane curves we reduce a calculation of its invariants to calculation of the intersection of kernels of some derivations.
We show that the polynomial entropy of homeomorphisms on regular curves is bounded above by one. Moreover, the polynomial entropy equals one under the fairly mild condition that the homeomorphism possesses a wandering point. We obtain a…