Endre Makai: A térfogatszorzat probléma a síkon 
(The volume product problem in the plane) 

Work joint with K. J. B\"or\"oczky.

Let $K \subset {\Bbb{R}}^d$ be a convex body containing $0$ in its
interior. Then its polar is defined by 
$K^*:=\{ y\in {\Bbb{R}}^d \mid\forall x \in K \,\, \langle x,y \rangle \le 1$. 
Then $K^* \subset {\Bbb{R}}^d$ also is a convex body containing $0$ in its 
interior. The question that we
investigate is the following: given the volume of $K$, what are the minimal
and maximal values of the volume of $K^*$. Equivalently: what are the minimal
and maximal values of $V(K)V(K^*)$, where $V$ means volume. (More exactly,
this formulation holds for $K$ $0$-symmetric only. For the general case a
slight modification is needed.) We solve the panar case for $K$ an $n$-gon:
the maximum is attained for the affine regular polygons. We determine in the
plane the maximum of the volume (area) product if we know $V(K)$ and either
the inradius or the circumradius of $K$, for the case when $K$ has $k$-fold
rotational symmetry about $0$, for $k \ge 3$.