## 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

Zbl 1437.52006
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