Let \(K\) be a convex body in \(\mathbb{R} ^n\) with the barycenter at the origin and \(| K| \) denote its volume. Let \(B_2^n\) and \((\cdot , \cdot )\) denote the canonical Euclidean ball and the canonical inner product on \(\mathbb{R} ^n\). Given a function \(f: K \to \mathbb{R}\) denote \(\| f\| _p = (\int _K | f(x)| ^p dx / | K| ) ^{1/p}\) and \(\| f\| _{\Psi} = \inf \{ t>0 : \int _K \exp (| f(x)| ^2/t^2) dx / | K| \leq 2\}\). We say that \(K\) is in the isotropic position if \(| K| =1\) and for every unit vector \(u\) one has \(\int _K (x, u) ^2 dx = L_K^2\) for some positive constant \(L_K\).
It is known that every convex body with the barycenter at the origin has a (unique up to rotations) linear image which is in the isotropic position. Moreover, one of the most interesting open problems of Asymptotic Theory, equivalent to the Slicing Problem, is to estimate \(L_K\), i.e. to find \(\sup L_K\) over all isotropic convex bodies \(K\subset \mathbb{R} ^n\) (see V.D. Milman and A. Pajor [Lect. Notes Math. 1376, 64–104 (1989; Zbl 0679.46012)] for the definitions and related discussion, and J. Bourgain [Lect. Notes Math. 1469, 127–137 (1991; Zbl 0773.46013)], S. Dar [Isr. J. Math. 97, 151–156 (1997; Zbl 0871.52007)], G. Paouris [Lect. Notes Math. 1745, 239–243 (2000; Zbl 0989.52003)] for the best known estimates).
The author proves two theorems connected to this problem. The first says that if \(K\) has a linear image \(K_1\) such that \(| K_1| =1\) and \(K_1 \subset a \sqrt{n} B_2^n\) (i.e. if the outer volume ratio of \(K\) is bounded) then there exists a unit vector \(u\) such that \(\| (\cdot , u) \| _{\Psi} \leq c a \| (\cdot , u) \| _1 \), where \(c\) is an absolute positive constant.
Note that every zonoid satisfies the assumption of the theorem.
The theorem is also related to the result of S. Alesker [Oper. Theory, Adv. Appl. 77, 1–4 (1995; Zbl 0834.52004)] and to the recent result of F. L. Nazarov and S. G. Bobkov [Lect. Notes Math. 1807, 53–69 (2003; Zbl 1039.52004)].
The second theorem states that there exists an absolute positive constant \(c\) such that if \(K\) is in the isotropic position and if \(\| (\cdot , u) \| _{\Psi} \leq C \| (\cdot , u) \| _1 \) for every \(u\) then \(K \subset c C^2 \sqrt{n} \ln n B_2^n\) (i.e. the outer volume ratio of \(K\) is bounded up to \(C ^2 \ln n\)).
For the entire collection see [Zbl 1009.00019].