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Efrén Morales-Amaya

Publications and source records attributed to Efrén Morales-Amaya.

9 recordsLinked to original sources
arXiv2026

A characterization of the ellipsoid in terms of pairs of sections associated by a harmonic homology

Let $K$ be a convex body in an affine chart of the $n$ dimensional real Projective space $\mathbb{RP}^n$, $n \geq 3$, let $H$ be a hyperplane which is not a support hyperplane of $K$ and let $p_1,p_2 \in \mathbb{RP}^n \setminus H$ be two distinct interior points of $K$. In this work we prove that if for every $(n-2)$-plane $l \subset H$, there exists a harmonic homology, with plane $G$ and center $τ$, such that $l\subset G$, $τ\in H$ and which maps the hypersection of $K$ defined by aff$\{p_1, l\}$ onto the hypersection of $K$ defined by aff$\{p_2, l\}$, then $K$ is an ellipsoid.

math.MG↗
arXiv2026

A characterization of the sphere in terms of the stereographic projection

Let $K$ be a convex body in the 3-dimensional Euclidian space $\mathbb{E}^3$ and let $N,S$ in the boubdary bd$K$ of $K$, $N\not=S$. Suppose that the support plane $Π_S$ of $K$ at $S$ is unique. For every point $x$ in bd$ K$, different than $N$, we define the stereographic projection $Ψ:\textrm{bd}K\backslash \{N\} \rightarrow Π_S$ of $x$ onto $Π_S$ as the point $y:=L(N,x)\cap Π_S$. It is a well known property of the sphere $\mathbb{S}^2$ in $\mathbb{E}^3$ that the stereographic projection maps circles onto circles (see \cite{Hilbert} pag. 248). In this work we investigate what geometric elements determines that this property is fulfilled. Here we demonstrate that the following two properties of a convex body $K\subset \mathbb{E}^3$ in terms of the stereographic projection characterize the sphere in $\mathbb{E}^3$: (1) The cones defined by the sections of $K$ and the point $N$ are axially symmetric (that is, they are invariant under a rotation by an angle of $π$). (2) given a section $K_Γ$ of $K$, the rotation that leaves the cone defined by $K_Γ$ and $N$ invariant is such that it maps $K_Γ$ into a homothetic figure to $Ψ(K_Γ)$ by a homothety with center of homothety at $N$. An important element in the proof of the main theorem of this work is a cha\-racterization of the circle based on a geometric property, which will be called the stereographic property. It is worth highlighting that the stereographic projection defined on the sphere maps circles onto circles is intimately linked to the conditions (1) and (2) and the stereographic property of the circle.

math.MG↗
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.

math.MG↗
arXiv2025

Convex bodies with sections with hyperplanes of symmetry

Let $K\subset \mathbb{R}^n$ be a convex body and let $p$ in the interior of $ K$, $n \geq 3$. The point $p$ is said to be a \textit{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, in addition, for every hyperplane $Π$ passing through $p$, the section $Π\cap K$ has a $(n-2)$-plane of symmetry which contains $p$, then the point $p$ is called a revolution point. In this work we prove that if for the convex body $K$, $n \geq 3$, there exists a hyperplane $H$, a point $p$ such that $p$ is a Larman point of $K$ but not a revolution point and, for every hyperplane $Π$ passing though $p$, the section $Π\cap K$ has an $(n-2)$-plane of symmetry parallel to $H$, then $K$ is an ellipsoid of revolution with an axis perpendicular to $H$.

math.MG↗
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 $D$.

math.MG↗
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.

math.MG↗
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.

math.MG↗
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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