scholarly journals Circle Numbers for Star Discs

ISRN Geometry ◽  
2011 ◽  
Vol 2011 ◽  
pp. 1-16 ◽  
Author(s):  
W.-D. Richter
Keyword(s):  
Star Bodies ◽  
Generalized Circle

The notion of a generalized circle number which has recently been discussed for -circles and ellipses will be extended here for star bodies and a class of unbounded star discs.

scholarly journals Some inequalities for star duality of the radial Blaschke-Minkowski homomorphisms

2020 ◽  
Vol 18 (1) ◽  
pp. 1064-1075
Author(s):  
Xia Zhao ◽  
Weidong Wang ◽  
Youjiang Lin
Keyword(s):  
Negative Answer ◽  
Positive Form ◽  
Dual Quermassintegrals ◽  
Monotonic Inequality ◽  
Star Bodies

Abstract In 2006, Schuster introduced the radial Blaschke-Minkowski homomorphisms. In this article, associating with the star duality of star bodies and dual quermassintegrals, we establish Brunn-Minkowski inequalities and monotonic inequality for the radial Blaschke-Minkowski homomorphisms. In addition, we consider its Shephard-type problems and give a positive form and a negative answer, respectively.

scholarly journals KHINTCHINE'S THEOREM AND TRANSFERENCE PRINCIPLE FOR STAR BODIES

2006 ◽  
Vol 02 (03) ◽  
pp. 431-453
Author(s):  
M. M. DODSON ◽  
S. KRISTENSEN
Keyword(s):  
Distance Function ◽  
Diophantine Approximation ◽  
Direct Proof ◽  
Distance Functions ◽  
Transference Principle ◽  
Simultaneous Diophantine Approximation ◽  
Star Bodies

Analogues of Khintchine's Theorem in simultaneous Diophantine approximation in the plane are proved with the classical height replaced by fairly general planar distance functions or equivalently star bodies. Khintchine's transference principle is discussed for distance functions and a direct proof for the multiplicative version is given. A transference principle is also established for a different distance function.

On the Dual p-Measures of Asymmetry for Star Bodies

2018 ◽  
Vol 23 (6) ◽  
pp. 465-470
Author(s):  
Xing Huang ◽  
Huawei Zhu ◽  
Qi Guo
Keyword(s):  
Measures Of Asymmetry ◽  
Star Bodies

scholarly journals General mixed chord-integrals of star bodies

2016 ◽  
Vol 46 (5) ◽  
pp. 1499-1518 ◽  
Author(s):  
Yibin Feng ◽  
Weidong Wang
Keyword(s):  
Star Bodies

Orlicz affine isoperimetric inequalities for star bodies

2022 ◽  
Vol 134 ◽  
pp. 102308
Author(s):  
Youjiang Lin ◽  
Dongmeng Xi
Keyword(s):  
Isoperimetric Inequalities ◽  
Star Bodies

STAR BODIES, STAR BIOS:

2015 ◽  
pp. 191-242
Keyword(s):  
Star Bodies

Orlicz Addition for Measures and an Optimization Problem for the -divergence

2019 ◽  
Vol 72 (2) ◽  
pp. 455-479
Author(s):  
Shaoxiong Hou ◽  
Deping Ye
Keyword(s):  
Optimization Problem ◽  
Jensen’S Inequality ◽  
Minkowski Inequality ◽  
Isoperimetric Inequalities ◽  
Jensen's Inequality ◽  
Dual Functional ◽  
Two Measures ◽  
Star Bodies

AbstractThis paper provides a functional analogue of the recently initiated dual Orlicz–Brunn–Minkowski theory for star bodies. We first propose the Orlicz addition of measures, and establish the dual functional Orlicz–Brunn–Minkowski inequality. Based on a family of linear Orlicz additions of two measures, we provide an interpretation for the famous $f$-divergence. Jensen’s inequality for integrals is also proved to be equivalent to the newly established dual functional Orlicz–Brunn–Minkowski inequality. An optimization problem for the $f$-divergence is proposed, and related functional affine isoperimetric inequalities are established.

The reduction of star sets

1952 ◽  
Vol 245 (892) ◽  
pp. 59-93 ◽  
Keyword(s):  
Finite Number ◽  
Algebraic Surfaces ◽  
Critical Determinant ◽  
Star Bodies

Mahler’s theory of irreducible star bodies is redeveloped and extended in a modified form. It is shown that any closed bounded star set S contains a closed irreducible star set T having the same critical determinant. Further, it is shown that, if the first set S is bounded by a finite number of algebraic surfaces, then there will be an irreducible set T which is also bounded by a finite number of algebraic surfaces.

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