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Jesús Jerónimo-Castro

Publications and source records attributed to Jesús Jerónimo-Castro.

10 recordsLinked to original sources
arXiv2026

An equichordal characterization of the ellipsoid and the sphere

Let $K$ and $L$ be two convex bodies in $\mathbb R^n$, $n\geq 3$, with $L\subset \text{int}\, K$. In this paper we prove the following result: if every two parallel chords of $K$, supporting $L$ have the same length, then $K$ and $L$ are homothetic and concentric ellipsoids. We also prove a similar theorem when instead of parallel chords we consider concurrent chords. We may also replace, in both theorems, supporting chords of $L$ by supporting sections of constant width. In the last section we also prove similar theorems where we consider projections instead of sections.

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arXiv2025

A Characterization of the sphere and a body of revolution by means of Larman points

Let $K\subset \mathbb{R}^n$, $n\geq 3$, be a convex body. A point $p$ the interior of $K$ is said to be a Larman point of $K$ if for every hyperplane $Π$ passing through $p$ the section $Π\cap K$ has a $(n-2)$-plane of symmetry. If $p$ is a Larman point of $K$ and, in addition, for every section $Π\cap K$, $p$ is in the corresponding $(n-2)$-plane of symmetry, then we call $p$ a revolution point of $K$. We conjecture that if $K$ contains a Larman point which is not a revolution point, then $K$ is either an ellipsoid or a body of revolution. This generalizes a conjecture of K. Bezdek for convex bodies in $\mathbb{R}^3$ to $n \geq 4$. We prove several results related to the conjecture for strictly convex origin symmetric bodies. Namely, if $K \subset \mathbb{R}^n$ is a strictly convex origin symmetric body that contains a revolution point $p$ which is not the origin, then $K$ is a body of revolution. This generalizes the False Axis of Revolution Theorem. We also show that if $p$ is a Larman point of $K \subset \mathbb{R}^3$ and there exists a line $L$ such that $p\notin L$ and, for every plane $Π$ passing through $p$, the line of symmetry of the section $Π\cap K$ intersects $L$, then $K$ is a body of revolution (in some cases, we conclude that $K$ is a sphere). We obtain a similar result for projections of $K$. Additionally, for $K \subset \mathbb{R}^n$, $n \geq 4$, we show that if every hyperplane section or projection of $K$ is a body of revolution and $K$ has a unique diameter $D$, then $K$ is a body of revolution with axis $D$.

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arXiv2025

On characteristic properties of the ellipsoid in terms of circumscribed cones of a convex body

We strongly believe that in order to prove two important geometrical pro\-blems in convexity, namely, the G. Bianchi and P. Gruber's Conjecture \cite{bigru} and the J. A. Barker and D. G. Larman's Conjecture \cite{Barker}, it is necessary obtain new characteristic properties of the ellipsoid, which involves the notions defined in such problems. In this work we present a series of results which intent to be a progress in such direction: Let $L,K\subset \mathbb{R}^n$ be convex bodies, $n\geq 3$, and $L$ be a subset in the interior of $K$. Then each of the following conditions i), ii) and iii) implies that $L$ is an ellipsoid. i) $L$ is $O$-symmetric and, for every $x$ in the boundary of $K$, the support cone $S(L,x)$ is ellipsoidal. ii) there exists a point $p\in \mathbb{R}^n$ such that for every $x$ in the boundary of $K$, there exists a point $y$ in the boundary of $K$ and hyperplane $Π$, passing through $p$, such that \[ S(L,x)\cap S(L,y)=Π\cap \textrm{bd } K. \] iii) $K$ and $L$ are $O$-symmetric, every $x$ in the boundary of $K$ is a pole of $L$ and $Ω_x:=S(L,x)\cap S(L,-x)$ is contained in the interior of $K$. In the case ii), $K$ is also an ellipsoid and it is concentric with $L$. On the other hand, let $K\subset \mathbb{R}^n$ be a $O$-symmetric convex body, $n\geq 3$, and let $B$ in $\mathbb{R}^n$ be a ball with centre at $O$. We are going to prove that if $B$ is small enough and all the sections of $K$ given by planes tangent to $B$ are $(n-1)$-ellipsoids, then $K$ is an $n$-ellipsoid.

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arXiv2024

Ball characterizations in planes and spaces of constant curvature, I

Let us have in S^2, R^2 or H^2 a pair of convex bodies, for S^2 different from S^2, such that the intersections of any congruent copies of them are centrally symmetric. Then our bodies are congruent circles. If the intersections of any congruent copies of them are axially symmetric, then our bodies are circles. Let us have in S^2, R^2 or H^2 proper closed convex subsets K,L with interior points, such that the numbers of the connected components of the boundaries of K and L are finite. If the intersections of any congruent copies of K and L are centrally symmetric, then K and L are congruent circles, or, for R^2, parallel strips. We describe all pairs of such subsets K,L, whose any congruent copies have an intersection with axial symmetry. For S^2, R^2 and H^2 there are 1, 5 and 9 cases, resp. Let us have in S^d, R^d or H^d proper closed convex C^2_+ subsets K,L with interior points, such that all sufficiently small intersections of their congruent copies are symmetric w.r.t. a particular hyperplane. Then the boundary components of both K and L are congruent, and each of them is a sphere, a parasphere or a hypersphere. Let us have a pair of convex bodies in S^d, R^d or H^d, which have at any boundary points supporting spheres, for S^d of radius less than π/2. If the convex hull of the union of any congruent copies of these bodies is centrally symmetric, then our bodies are congruent balls, for S^d of radius less than π/2. An analogous statement holds for symmetry w.r.t. a particular hyperplane. For d=2 suppose the existence of the above supporting circles, for S^2 of radius less than π/2, and for S^2 smoothness of K and L. If we suppose axial symmetry of all the above convex hulls, then our bodies are circles, for S^2 of radii less than π/2.

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arXiv2021

Some results about equichordal convex bodies

Let $K$ and $L$ be two convex bodies in $\mathbb R^n$, $n\geq 2$, with $L\subset \text{int}\, K$. We say that $L$ is an equichordal body for $K$ if every chord of $K$ tangent to $L$ has length equal to a given fixed value $λ$. J. Barker and D. Larman proved that if $L$ is a ball, then $K$ is a ball concentric with $L$. In this paper we prove that there exist an infinite number of closed curves, different from circles, which possess an equichordal convex body. If the dimension of the space is more than or equal to 3, then only Euclidean balls possess an equichordal convex body. We also prove some results about isoptic curves and give relations between isoptic curves and convex rotors in the plane.

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arXiv2021

Convex bodies with equipotential circles

Given a convex body $K\subset \mathbb R^2$ we say that a circle $Ω\subset \text{int} \ K$ is an equipotential circle if every tangent line of $Ω$ cuts a chord $AB$ in $K$ such that for the contact point $P=Ω\cap AB$ it holds that $|AP|\cdot|PB|=λ$, for a suitable constant number $λ$. The main result in this article is the following: Let $K\subset\mathbb R^2$ be a convex body which has an equipotential circle $\mathcal B$ with centre $O$ in its interior. Then $K$ has centre of symmetry at $O$, moreover, if none chord of $K$ which is tangent to $\mathcal B$ subtends an angle $π/2$ from $O$, then $K$ is a disc. We also derive some results which characterizes the ellipsoid and the sphere in $\mathbb R^3$ and introduce also the concept of equireciprocal disc.

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arXiv2015

On a problem by Dol'nikov

In 2011 at an Oberwolfach workshop in Discrete Geometry, V. Dol'nikov posed the following problem. Consider three non-empty families of translates of a convex compact set $K$ in the plane. Suppose that every two translates from different families have a point of intersection. Is it always true that one of the families can be pierced by a set of three points? A result by R. N. Karasev from 2000 gives, in fact, an affirmative answer to the "monochromatic" version of the problem above. That is, if all the three families in the problem coincide. In the present paper we solve Dol'nikov's problem positively if $K$ is either centrally symmetric or a triangle, and show that the conclusion can be strengthened if $K$ is an euclidean disk. We also confirm the conjecture if we are given four families satisfying the conditions above.

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arXiv2013

Two characterizations of ellipsoidal cones

We give two characterizations of cones over ellipsoids. Let $C$ be a closed pointed convex linear cone in a finite-dimensional real vector space. We show that $C$ is a cone over an ellipsoid if and only if the affine span of $\partial C \cap \partial(a - C)$ has dimension $\dim(C) - 1$ for every point $a$ in the relative interior of $C$. We also show that $C$ is a cone over an ellipsoid if and only if every bounded section of $C$ by an affine hyperplane is centrally symmetric.

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