Mihály Bencze

The aim of this paper is to investigate when a linear normed space is an inner product space. Several conditions in a linear normed space are formulated with the help of inequalities. Some of them are from the literature and others are new. We prove that these conditions are equivalent with the fact that the norm is induced by an inner product. One of the new results is the following: in an inner product space, the sum of opposite edges of a tetrahedron are the sides of an acute angled triangle. The converse of this result holds also. More precisely, this property characterizes inner product spaces. Another new result is the following: in a tetrahedron, the sum of squares of opposite edges are the lengths of a triangle. We prove also that this property characterizes inner product spaces. In addition, we give simpler proofs to some theorems already known from the publications of other authors.

In the paper, our main aim is to generalize the q ^th dual volume to L _p space, and introduce pq ^th - dual mixed volume by calculating the first order variation of q ^th dual volumes. We establish the L _pq -Minkowski inequality for pq ^th -dual mixed volumes and L _pq -Brunn-Minkowski inequality for the q ^th -dual volumes, respectively. The new inequalities in special case yield some new dual inequalities for the q ^th -dual volumes.

In this paper we prove some results concerning bicentric quadrilaterals. We offer a new proof of the Blundon-Eddy inequality, which we use to obtain other inequalities in bicentric quadrilaterals.