Asymptotic theory of finite dimensional normed spaces
Asymptotic theory of finite dimensional normed spaces
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For an arbitrary n-dimensional convex body, at least almost n Steiner symmetrizations are required in order to symmetrize the body into an isomorphic ellipsoid. We say that a body $T \subset \mathbb{R}^n$ is ‘quickly symmetrizable with function $c(\varepsilon)$’ if for any $\varepsilon 0$ there exist only $\lfloor \varepsilon n \rfloor$ symmetrizations that transform T into a body which is $c(\varepsilon)$-isomorphic to an ellipsoid. In this note we ask, given a body $K \subset \mathbb{R}^n$, whether it is possible to remove a small portion of its volume and obtain a body $T \subset K$ which is quickly symmetrizable. We show that this question, for $c(\varepsilon)$ polynomially depending on $\frac{1}{\varepsilon}$, is equivalent to the slicing problem.