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Stronger versions of the Orlicz-Petty projection inequality. (English) Zbl 1291.52012

Given a convex body \(K\) containing the origin in its interior, and a convex function \(\varphi:{\mathbb R}\longrightarrow[0,\infty)\) with \(\varphi(0)=0\) and \(\widetilde{\varphi}(t)=\varphi(t)+\varphi(-t)>0\) for \(t\neq 0\), the Orlicz projection body \(\Pi_{\varphi}K\) is the convex body whose support function is given by \[ h_{\Pi_{\varphi}K}(x)=\min\left\{\lambda>0:\int_{S^{n-1}}\varphi\left(\frac{x\cdot w}{\lambda h_K(w)}\right)h_K(w)dS_K(w)\leq nV(K)\right\}, \] where \(V(\cdot)\) denotes the volume and \(S_K\) is the surface area measure of \(K\) on \(S^{n-1}\). In [Adv. Math. 223, No. 1, 220–242 (2010; Zbl 1437.52006)], E. Lutwak et al. obtained the so-called Orlicz-Petty projection inequality, namely, that the volume ratio \(V(\Pi^*_{\varphi}K)/V(K)\) is maximized when \(K\) is a \(o\)-symmetric ellipsoid. Here, \(\Pi^*_{\varphi}K\) denotes the polar body of \(\Pi_{\varphi}K\). They proved, moreover, that if \(\varphi\) is strictly convex, then the only maximizers are the symmetric ellipsoids, and conjectured that this is the case for any \(\varphi\).
In the paper under review, the author confirms the above conjecture, showing also a stability version of the Orlicz-Petty projection inequality. More precisely, he proves that \[ \frac{V(\Pi^*_{\varphi}K)}{V(K)}\leq \left(1-\gamma_{n,\varphi}\delta^{840n}\widetilde{\varphi}(\delta^{840})\right) \frac{V(\Pi^*_{\varphi}B^n)}{V(B^n)}, \] where \(\delta=\delta_{BM}(K,B^n)\) is the Banach-Mazur distance of \(K\) and the Euclidean unit ball \(B^n\), and \(\gamma_{n,\varphi}>0\) only depends on \(n\) and \(\varphi\). Another stability result using a different distance and for particular functions \(\varphi\) is also obtained.

MSC:

52A40 Inequalities and extremum problems involving convexity in convex geometry

Citations:

Zbl 1437.52006
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