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Volume inequalities for asymmetric Orlicz zonotopes. (English) Zbl 1405.52004

The authors extend the asymmetric \(L_p\) volume product and asymmetric \(L_p\) volume ratio inequalities due to M. Weberndorfer [Adv. Math. 240, 613–635 (2013; Zbl 1296.52008)] to the Orlicz setting. The proof here uses shadow system, which is the same as in the Weberndorfer paper [loc. cit.].
Let \(\phi: [0,\infty)\rightarrow [0,\infty)\) is a convex, strictly increasing function satisfying \(\phi(0)=0\) and \(\phi(1)=1\). Suppose \(\Lambda\) be a finite set of non-zero vectors spanning \(\mathbb{R}^n\). The asymmetric Orlicz zonotope \(Z_\phi^+ \Lambda\) is defined to be the convex body whose support function is given by \[ h_{Z_\phi^+ \Lambda}(u) = \inf\left\{\lambda>0: \sum_{w\in \Lambda} \phi\left(\frac{\langle w, u\rangle_+}{\lambda}\right)\leq 1\right\}, \] for each \(u\in \mathbb{R}^n\). Denote by \(Z_\phi^{+,*}\Lambda\) the polar body of \(Z_\phi^+ \Lambda\) with respect to its Santal point, and \(\Lambda_\perp\) be a canonical basis of \(\mathbb{R}^n\). The authors show the following volume product inequality: \[ V(Z_\phi^{+,*}\Lambda)V(Z_1^+\Lambda)\geq V(Z_\phi^{+,*}\Lambda_\perp)V(Z_1^+\Lambda_\perp) \] and the following volume ratio inequality: \[ \frac{V(Z_\phi^+ \Lambda)}{V(Z_1^+\Lambda)}\leq \frac{V(Z_\phi^+ \Lambda_\perp)}{V(Z_1^+\Lambda_\perp)}. \] Equality conditions for the above two inequalities are also completely characterized.

MSC:

52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)
52A40 Inequalities and extremum problems involving convexity in convex geometry
52A38 Length, area, volume and convex sets (aspects of convex geometry)

Citations:

Zbl 1296.52008
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References:

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