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

• DOI: 10.48550/arxiv.2401.03983
• Publication Date: 2024-01-01
• Authors: E. Morales‐Amaya; Geronimmo Mondragón; Jesús Jerónimo-Ca...
• Keywords: Ellipsoid; Hyperplane; Convexity

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