Variations and generalizations of several classical theorems concerning characterizations ellipsoids are developed. In particular, these lead to a short comprehensible proof the false centre theorem.

By introducing the concept of polarity in convex sets, it is possible, a natural way, to generalize several classic characterizations ellipsoids, showing that all them depend upon and are related projective centre symmetry. Using these ideas, also possible develop new ellipsoids propose problems.

Let K be a convex body and let p0∈int K. Suppose that in every direction we can choose continuously section of which is translated copy the corresponding parallel through p0. Our main result essentially claims if all these pairs sections are different almost everywhere, then an ellipsoid.

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),...

Generalizing a Theorem due to A. Rogers \cite{ro1}, we are going prove that if for pair of convex bodies $K_{1},K_{2}\subset \Rn$, $n\geq 3$, there exists hyperplane $H$ and different points $p_1$ $p_2$ in $\Rn \backslash H$ such each $(n-2)$-plane $M\subset H$, \textit{mirror} which maps the hypersection $K_1$ defined by $\aff\{ p_1,M\}$ onto $K_2$ p_2,M\}$, then $K_2$.

Abstract Let n ≥ 3 and let K ⊂ ℝ be a convex body. A point p ∈ int is said to Larman of if for every hyperplane Π passing through , the section ∩ has an ( – 2)-plane symmetry. If in corresponding symmetry, then we call revolution . We conjecture that contains which not point, either ellipsoid or body revolution. This generalizes Bezdek = 3. prove several results related strictly origin symmetric bodies. Namely, origin, False Axis Revolution Theorem [7]. also show there exists line L such ∉...

In this work we present a theorem regarding two convex bodies $K_1, K_2\subset \mathbb{R}^{n}$, $n\geq 3$, and families of sections them, given by tangent planes spheres $S_i\subset \textrm{int}\textrm{ } K_i$, $i=1,2$ such that, for every pair $\Pi_1$, $\Pi_2$ parallel supporting $S_1$, $S_2$, respectively, which are corresponding (this means, that the outer normal vectors half spaces determined have same direction), $\Pi_1\cap K_1$, $\Pi_2\cap K_2$ translated, claims if $S_2$ radius,...

Given a convex body K⊂R2 we say that circle B⊂intK is an equipotential if variable chord AB of K tangent to B at P has the product |AP|·|PB| constant. The main result this article following: Let be interior centered O. Then center symmetry O, moreover, no subtends angle π/2 from then disk.

L. Fejes Tóth [Acta Math. Acad. Sci. Hungar, 13: 379–382, 1962] introduced the notion of fixing system for a compact, convex body M ⊂ Rn. Such F bd stabilizes with respect to translations. In particular, every minimal is primitive, i.e., no proper subset system. [Boltyanski and Martini, Combinatorial geometry belt bodies] lower upper bounds cardinalities mimimal systems are indicated. Here we give an improved bound show by examples, now both exact. Finally, formulate Problem.

Abstract We shall prove the following shaken Rogers's theorem for homothetic sections: Let K and L be strictly convex bodies suppose that every plane H through origin we can choose continuously sections of , parallel to which are directly homothetic. Then

Abstract We prove the following result: Let K be a strictly convex body in Euclidean space ℝ n , ≥ 3, and let L hypersurface which is image of an embedding sphere 𝕊 –1 such that contained interior . Suppose that, for every x ∈ there exists y support cones with apexes at differ by central symmetry. Then are centrally symmetric concentric.

Abstract In this paper we proved the following: Let be two O ‐symmetric convex bodies with strictly convex. Suppose that from every x in graze is a planar curve and K almost free respect to L . Then an ellipsoid.

Let $K\subset \Rn$, $n\geq 3$, be a convex body. A point $p\in \Rn$ is said to \textit{Larman point} of $K$ if, for every hyperplane $\Pi$ passing through $p$, the section $\Pi\cap K$ has $(n-2)$-plane symmetry. If $p \in Larman and in addition, symmetry which contains then we call $p$ \textit{revolution $K$. In this work prove that if \Rt$ strictly centrally symmetric body with centre at $o$, there exists line $L$ such $p\notin L$ and, plane $\Gamma$ $\Gamma \cap intersects $L$, revolution...

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 parallel chords of $K$, supporting have same length, then are homothetic concentric ellipsoids. We also a similar theorem when instead consider concurrent chords. may replace, both theorems, by sections constant width. last section theorems where projections sections.

In this work we prove the following: let $K$ be a convex body in Euclidean space $\mathbb{R}^n$, $n\geq 3$, contained interior of unit ball and $p\in \mathbb{R}^n$ point such that, from each $\mathbb{S}^{n-1}$, looks centrally symmetric $p$ appears as center, then is ball.