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{We show in this paper that two normal elliptic sections through every point of the boundary of a smooth convex body essentially characterize an ellipsoid and furthermore, that four different pairwise non-tangent elliptic sections through…

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…

In this note we prove two ellipsoid characterization theorems. The first one is that if $K$ is a convex body in a normed space with unit ball $M$, and for any point $p \notin K$ and in any 2-dimensional plane $P$ intersecting $\inter K$ and…

We characterize all possible relative positions between a hyperboloid of one sheet and a sphere through the roots of a characteristic polynomial associated to these quadrics. The classification is also suitable for a hyperboloid and a…

One of the most important problems in Geometric Tomography is to establish properties of a given convex body if we know some properties over its sections or its projections. There are many interesting and deep results that provide…

Let $E_1,E_2\subset \mathbb{R}^n$ be two homothetic solid ellipsoids, $n\geq 3$, with center at the origin $O$ of a system coordinates of $\mathbb{R}^n$, and $E_1\subset E_2$. Then there exists a $O$-symmetric ellipsoid $E_3$ such that…

We show some characterizations of hyperspheres in the $(n+1)$-dimensional Euclidean space ${\Bbb E}^{n+1}$ with intrinsic and extrinsic properties such as the $n$-dimensional area of the sections cut off by hyperplanes, the…

We give two characterizations of cones over ellipsoids in real normed vector spaces. Let $C$ be a closed convex cone with nonempty interior such that $C$ has a bounded section of codimension $1$. We show that $C$ is a cone over an ellipsoid…

A hypersurface $M$ in $\mathbb{R}^n$, $n \geq 4$, has central ovaloid property if $M$ intersects some hyperplane transversally along an ovaloid and every such ovaloid on $M$ has central symmetry. We show that a complete, connected, smooth…

In this paper, we study locally strongly convex Tchebychev hypersurfaces, namely the {\it centroaffine totally umbilical hypersurfaces}, in the $(n+1)$-dimensional affine space $\mathbb{R}^{n+1}$. We first make an ordinary-looking…

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…

We establish some characterizations of elliptic hyperboloids (resp., ellipsoids) in the $(n+1)$-dimensional Euclidean space ${\Bbb E}^{n+1}$, using the $n$-dimensional area of the sections cut off by hyperplanes and the $(n+1)$-dimensional…

We find that for any n-dimensional, compact, convex subset K of R^{n+1} there is an affinely-spherical hypersurface M in R^{n+1} with center at the relative interior of K, such that the disjoint union of M and K is the boundary of an…

Liebmann's Theorem asserts that a compact, connected, convex surface with constant mean curvature (CMC) in the Euclidean space must be a totally umbilical sphere. In this article we extend Liebmann's result to hypersurfaces with boundary.…

We prove the following localized version of a classical ellipsoid characterization: Let $B\subset\mathbb R^3$ be convex body with a smooth strictly convex boundary and 0 in the interior, and suppose that there is an open set of planes…

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…

The spherical centroid body of a centrally-symmetric convex body in the Euclidean unit sphere is introduced. Two alternative definitions - one geometric, the other probabilistic in nature - are given and shown to lead to the same objects.…

In this work we prove the following result: Let $K$ be a strictly convex body in the Euclidean space $\mathbb{R}^n, n\geq 3$, and let $L$ be a hypersurface, which is the image of an embedding of the sphere $\mathbb{S}^{n-1}$, such that $K$…

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 $\Sigma(L,x)$ is a planar…

This paper is devoted to measures of symmetry based on distance between centroid and one of the centers of John and Lowner ellipsoid. The author proves the accuracy of the derived upper bounds for the considered measures of symmetry.