Efren Morales Amaya | Searcharxiv

arXiv2025

Characterization of the sphere by means of congruent support cones

Let $M$ be a convex body and let $K$ be a closed convex surface $K$ both contained in the Euclidean space $\mathbb{E}^3$. What can we say about $M$ if $K$ encloses $M$ and if from all the points in $K$ the body $M$ looks the same? In this work we are going to present a result which claims that if for every two support cones $C_x$, $C_y$ of $M$, with apexes $x,y \in K$, respectively, there exists $Φ$ in the semi direct product of the orthogonal group $O(3)$ and $\mathbb{E}^3$ such that $$C_y=Φ(C_x),$$ and this can be done in a continuous way, then $M$ is a sphere.

math.MG↗

arXiv2025

A characterization of the ellipsoid through planar grazes

In this paper we proved the following: \emph{Let $K, L\subset \mathbb R^3$ be two $O$-symmetric convex bodies with $L\subset \emph{int} K$ strictly convex. Suppose that from every $x$ in $\emph{bd} K$ the graze $Σ(L,x)$ is a planar curve and $K$ is almost free with respect to $L$. Then $L$ is an ellipsoid.}

math.MG↗