Artem Zvavitch
Kent State University
Title
Zonoids with zonoid polars need not approach the Euclidean ball
Abstract
Rolf Schneider showed, using perturbations of the Euclidean ball, that a zonoid whose polar is also a zonoid need not be an ellipsoid. This raises an asymptotic question: does the maximal Banach–Mazur distance from the Euclidean ball among such bodies tend to one as the dimension increases?
We give a negative answer. For every $n\ge 2$, we exhibit a zonoid $Z_n\subset\mathbb R^n$ whose polar is also a zonoid and whose exact Banach–Mazur distance from the Euclidean ball is independent of the dimension and greater than $1.09$. The construction uses a family of Gaussian zonoids studied by Leo Mathis, within the broader theory of Gaussian and lift zonoids. The main analytic step is a complete-monotonicity theorem that gives a representation of the polar support function as a positive mixture of Gaussian zonoid support functions.
The same representation allows us to extend the construction to infinite dimensions. We obtain a separable infinite-dimensional real Banach space $X$ such that both $X$ and $X^*$ embed linearly isometrically into $L_1$, although $X$ is not isometric to a Hilbert space. This answers negatively the real infinite-dimensional form of Grothendieck’s isometric question.
This is joint work with Dmitry Ryabogin.
