The minimal Orlicz mean width of convex bodies

ABSTRACT

In this paper, we generalize the minimal p-mean width of convex bodies (including the classic minimal mean width) to the Orlicz Brunn–Minkowski–Firey theory. The concept of Orlicz mean width of a convex body K in ${\mathbb{R}}^n$, $w_{\varphi }\left(K\right)$, is introduced. Then we study the minimization problems of the form $min\left\{w_{\varphi }\right(TK):T\in \mathrm{S}\mathrm{L}(n\left)\right\}$ and show that bodies which appear as solutions of such problems satisfy isotropic conditions of a suitable measure. Finally, the characteristics of the condition for the minimum $M_{\varphi }\left(K\right)M_{\varphi }^{\ast }\left(K\right)$ are obtained, a stability result for $L_p$-centroid bodies is established, and some new applications for the minimal Orlicz mean width position are provided.

Keywords:

• Convex body
• isotropic measure
• p-mean width
• Orlicz mean width

2010 Mathematics Subject Classifications:

• 52A30
• 52A40

Acknowledgements

Author would like to thank Dr. J. Li and Dr. Y. B. Feng for some helpful help.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work is supported by the National Natural Science Foundation of China (Grant No. 11561020).