Convex polytopes whose projection bodies and difference sets are polars
By means of analytic methods and combinatorial properties of hyperplane arrangements it is shown that a convex d-polytope P (d\(\geq 3)\) is a simplex if and only if its projection body \(\Pi\) P and its difference body DP are polars. For \(d=2\) this polarity holds precisely for those convex figures whose difference sets are bounded by Radon curves, for \(d\geq 3\) (besides simplices) obviously the ellipsoids have this correspondence. Thus, there remains the question whether simplices and ellipsoids are the only convex bodies in \({\mathbb{R}}^ d\) (d\(\geq 3)\) with this property. At least it is known that polarity of \(\Pi\) K and DK (with respect to a sphere of radius \(r\in {\mathbb{R}}^+)\) for an arbitrary convex body K of volume V(K) implies \[ (2\omega_{d-1}/\omega_ d)V(K)\leq r^ 2\leq d V(K),\quad \omega_ d:=\pi^{d/2}/\Gamma (1+d/2),\quad d\geq 2, \] with equality on the left hand side for ellipsoids and on the right hand side if and only if K is a simplex.
scientific article; zbMATH DE number 4113411
scientific article