The Characteristic Properties of the Minimal Lp-Mean Width

Abstract

Giannopoulos proved that a smooth convex body has minimal mean width position if and only if the measure ℎ ( ) (d ), supported on −1 , is isotropic. Further, Yuan and Leng extended the minimal mean width to the minimal -mean width and characterized the minimal position of convex bodies in terms of isotropicity of a suitable measure. In this paper, we study the minimal -mean width of convex bodies and prove the existence and uniqueness of the minimal -mean width in its SL( ) images. In addition, we establish a characterization of the minimal -mean width, conclude the average ( ) with a variation of the minimal -mean width position, and give the condition for the minimum position of ( ).

Strengthened inequalities for the mean width and the ℓ‐norm

“…Results similar to Theorem 1.2 are proved by Ma [55] in the L p setting. The main goal of the present paper is to provide stronger stability versions of Theorems 1.1 and 1.2.…”

“…Applying first (54), using ( 57) and ( 58) in ( 56), then the substitution z = Θ(x), and finally also (55), we get…”

Strengthened inequalities for the mean width and the $\ell$-norm

“…Results similar to Theorem 1.2 are proved by Ma [56] in the L p setting. The main goal of the present paper is to provide stronger stability versions of Theorem 1.1 and Theorem 1.2.…”