The generalized \(L_p\)-mixed affine surface area
type: Item
instance of: scholarly article
title: The generalized \(L_p\)-mixed affine surface area
zbMATH Open document ID: 1352.52011
author: Tongyi Ma
publication date: 2015-11-06T00:00:00Z
review text: The concept of classical affine surface area was generalized in many ways, in the sense meant here mainly by E. Lutwak (from mixed affine surface area over \(L_p\)-affine surface area to \(L_p\)-mixed affine surface area). Extensions to \(i\)th \(L_p\)-mixed affine surface area followed, where \(i\) is any real number. In the present article, the author studies so-called \((i,j)\)-type \(L_p\)-mixed affine surface area, having Lutwak's notions as subcases. Also, the author lays special emphasize on the case of \((i,-p)\)-type \(L_p\)-mixed affine surface area, establishing the Minkowski inequality and the \(L_p\)-Petty affine projection inequality for this case. Further deep results are obtained, among them also an affirmative answer for the generalized \(L_p\)-Winterniz monotonicity problem.
reviewed by: Horst Martini
Mathematics Subject Classification ID: 52A40
Mathematics Subject Classification ID: 52A20
zbMATH DE Number: 6505623
zbMATH Keywords: convex body
zbMATH Keywords: star body
zbMATH Keywords: \(i\)th \(L_p\)-mixed curvature function
zbMATH Keywords: \(i\)th \(L_p\)-mixed curvature image
zbMATH Keywords: \((i,j)\)-type \(L_p\)-mixed affine surface area
MaRDI profile type: Publication
full work available at URL: s10114-015-3140-0
OpenAlex ID: W2195011574
cites work: Q4856168
cites work: Q3245768
cites work: Zur Affinoberfläche konvexer Körper. (On affine surfaces of convex bodies)
cites work: Über einige Eigenschaften der Affinoberfläche beliebiger konvexer Körper. (On some properties of the affine surface area of arbitrary convex bodies)
cites work: Bemerkungen zur Definition einer erweiterten Affinoberfläche von E. Lutwak. (Remarks on the definition of extended affine surface area given by E. Lutwak)
cites work: On the history of the affine surface area for convex bodies
cites work: Q4377597
cites work: Q4778514
cites work: A characterization of affine surface area
cites work: Q3697820
cites work: On some affine isoperimetric inequalities
cites work: Mixed affine surface area
cites work: Centroid Bodies and Dual Mixed Volumes
cites work: Extended affine surface area
cites work: The Brunn-Minkowski-Firey theory. II: Affine and geominimal surface areas
cites work: The Brunn-Minkowski-Firey theory. I: Mixed volumes and the Minkowski problem
cites work: Dual mixed volumes
cites work: Inequalities for Hadwiger's harmonic quermassintegrals
cites work: \(L_ p\) affine isoperimetric inequalities.
cites work: Mixed Projection Inequalities
cites work: The \(L_p\)-curvature images of convex bodies and \(L_p\)-projection bodies
cites work: FOURIER TRANSFORM AND Lp-MIXED PROJECTION BODIES
cites work: Inequalities for \(L_p\)-mixed curvature images
cites work: On the analog of Shephard problem for the L_p-projection body
cites work: Q4926794
cites work: On the \(p\)-affine surface area
cites work: Geominimal surface area
cites work: Q5657318
cites work: Q3697700
cites work: Q5287551
cites work: On the Affine Surface Area
cites work: Surface bodies and \(p\)-affine surface area
cites work: Illumination bodies and affine surface area
cites work: \(L^{p}\) solutions of BSDEs with stochastic Lipschitz condition
cites work: \(L_p\)-dual mixed quermassintegrals
cites work: The Petty projection inequality for \(L_p\)-mixed projection bodies
cites work: Q3593803
DOI: 10.1007/S10114-015-3140-0
Recommended article: L_p-mixed affine surface area
Recommended article: \(L_{p}\)-mixed affine surface area
Recommended article: \(L_p\)-dual mixed affine surface areas
Recommended article: On the p-mixed affine surface area
Recommended article: The mixed Lp affine surface areas for functions
Recommended article: On $L_p$-affine surface areas
Recommended article: Some inequalities for \(L_p\)-mixed affine surface area
Recommended article: The generalized \(L_p\)-affine surface areas and Shephard problem
Recommended article: \(L_p\)-dual affine surface area
published in: Acta Mathematica Sinica, English Series
description: scientific article; zbMATH DE number 6505623