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

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

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 $\Pi$, passing through $p$, such that \[ S(L,x)\cap S(L,y)=\Pi \cap \textrm{bd } K. \] iii) $K$ and $L$ are $O$-symmetric, every $x$ in the boundary of $K$ is a pole of $L$ and $\Omega_x:=S(L,x)\cap S(L,-x)$ 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.

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.

Let $K\subset \Rn$, $n\geq 3$, be a convex body. A point $p\in \Rn$ is said to be a \textit{Larman point} of $K$ if, for every hyperplane $\Pi$ passing through $p$, the section $\Pi\cap K$ has a $(n-2)$-plane of symmetry. If a point $p \in \Rn$ is a Larman point and if, in addition, for every hyperplane $\Pi$ passing through $p$, the section $\Pi\cap K$ has a $(n-2)$-plane of symmetry which contains $p$, then we call $p$ a \textit{revolution point} of $K$. In this work we prove that if $K\subset \Rt$ is a strictly convex centrally symmetric body with centre at $o$, $p$ is a Larman point of $K$ and there exists a line $L$ such that $p\notin L$ and, for every plane $\Gamma$ passing through $p$, the section $\Gamma \cap K$ has a line of symmetry which intersects $L$, then $K$ is a body of revolution (in some cases, we conclude that $K$ is a sphere). On the other hand, we also prove that if $p$ is a revolution point such that $p\not=o$, then $K$ is a body of revolution.

We present in this work a new and simple proof of the false centre theorem.

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 $\lambda$. 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.

Given a convex body $K\subset \mathbb R^2$ we say that a circle $\Omega\subset \text{int} \ K$ is an equipotential circle if every tangent line of $\Omega$ cuts a chord $AB$ in $K$ such that for the contact point $P=\Omega\cap AB$ it holds that $|AP|\cdot|PB|=\lambda$, for a suitable constant number $\lambda$. 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 $\pi/2$ 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.