To prove two important geometrical problems in convexity, namely, the Conjecture of Bianchi and Gruber (Arch Math 49:344–350, 1987) and the Conjecture of Barker and Larman (Discrete Math 241:79–96, 2001), it is necessary to 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 intend to be a progress in such direction: Let \(L,K\subset \mathbb {R}^{n}\) be convex bodies, \(n\ge 3\) , and \(L\subset \textrm{int}K\) . Then, each of the following conditions (i), (ii) and (iii) implies that \(L\) is an ellipsoid. (i) \(L\) is \(O\) -symmetric and, for every \(x\in \textrm{bd}K\) , the support cone \(S(L,x)\) is ellipsoidal.
(ii) there exists a point \(p\in \mathbb {R}^{n}\) , such that for every \(x\in \textrm{bd}K\) , there exists \(y\in \textrm{bd}K\) and hyperplane \(\Pi \) , passing through \(p\) , such that \(\begin{aligned} S(L,x)\cap S(L,y)=\Pi \cap \textrm{bd}K. \end{aligned}\)
(iii) \(K\) and \(L\) are \(O\) -symmetric, every \(x\in \textrm{bd}K\) is a pole of \(L\) and \(\Omega _x:=S(L,x)\cap S(L,-x)\) is contained in \(\textrm{int}K\) .
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 an \(O\) -symmetric convex body, \(n\ge 3\) , and let \(B\subset \textrm{int}\mathbb {R}^{n}\) be a ball with center 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.
