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An application of the Fourier transform to sections of star bodies. (English) Zbl 0916.52002

Let \(K\) be a centrally symmetric star body in Euclidean space \(\mathbb{R}^n\), and let \(\| x\|:= \min\{a>0: x\in aK\}\).
It is proved that \[ \text{Vol}_{n- 1}(K\cap \xi^\perp)= {1\over \pi(n- 1)} (\| x\|^{-n+1})^{\wedge}(\xi) \] for unit vectors \(\xi\), where \(\xi^\perp\) is the hyperplane through \(0\) orthogonal to \(\xi\) and \(\wedge\) denotes the Fourier transform. The result is then used to show that, for the unit ball \(B_p\) of \(\ell^n_p\), \(p\in(0,2)\), the section volume \(\text{Vol}_{n- 1}(B_p\cap \xi^\perp)\) is minimal if \(\xi\) has the direction of the vector \((1,1,\dots, 1)\). This had been conjectured by Meyer and Pajor.

MSC:

52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)
52A21 Convexity and finite-dimensional Banach spaces (including special norms, zonoids, etc.) (aspects of convex geometry)
42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
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