Successive radii and Minkowski addition

2010 ◽  
Vol 166 (3-4) ◽  
pp. 395-409 ◽  
Author(s):  
Bernardo González ◽  
María A. Hernández Cifre
Keyword(s):  
Minkowski Addition

scholarly journals Continued fractions built from convex sets and convex functions

2015 ◽  
Vol 17 (05) ◽  
pp. 1550003 ◽  
Author(s):  
Ilya Molchanov
Keyword(s):  
Convex Functions ◽  
Continued Fractions ◽  
Convex Sets ◽  
Sufficient Conditions ◽  
Ordered Semigroup ◽  
Minkowski Addition ◽  
Partially Ordered ◽  
The Family ◽  
Sufficient Conditions For Convergence

In a partially ordered semigroup with the duality (or polarity) transform, it is possible to define a generalization of continued fractions. General sufficient conditions for convergence of continued fractions are provided. Two particular applications concern the cases of convex sets with the Minkowski addition and the polarity transform and the family of non-negative convex functions with the Legendre–Fenchel and Artstein-Avidan–Milman transforms.

Random compact convex sets which are infinitely divisible with respect to Minkowski addition

1979 ◽  
Vol 11 (4) ◽  
pp. 834-850 ◽  
Author(s):  
Shigeru Mase
Keyword(s):  
Convex Sets ◽  
Compact Convex ◽  
Infinite Divisibility ◽  
Spectral Measures ◽  
Infinitely Divisible ◽  
Minkowski Addition ◽  
Closed Sets ◽  
Random Closed Sets ◽  
Canonical Representations ◽  
Laplace Transformations

Random closed sets (in Matheron's sense) which are a.s. compact convex and contain the origin are considered. The totality of such random closed sets are closed under the Minkowski addition and we can define the concept of infinite divisibility with respect to Minkowski addition of random compact convex sets. Using a generalized notion of Laplace transformations we get Lévy-type canonical representations of infinitely divisible random compact convex sets. Isotropic and stable cases are also considered. Finally we get several mean formulas of Minkowski functionals of infinitely divisible random compact convex sets in terms of their Lévy spectral measures.

Algebraically inspired results on convex functions and bodies

2016 ◽  
Vol 18 (06) ◽  
pp. 1650027 ◽  
Author(s):  
Liran Rotem
Keyword(s):  
Convex Functions ◽  
Final Section ◽  
Polar Body ◽  
Convex Bodies ◽  
Type Inequality ◽  
Dual Function ◽  
Minkowski Addition ◽  
Geometric Means ◽  
Better Than

We show how algebraic identities, inequalities and constructions, which hold for numbers or matrices, often have analogs in the geometric classes of convex bodies or convex functions. By letting the polar body [Formula: see text] or the dual function [Formula: see text] play the role of the inverses “[Formula: see text]” and “[Formula: see text]”, we are able to conjecture many new results, which often turn out to be correct. As one example, we prove that for every convex function [Formula: see text] one has [Formula: see text] where [Formula: see text]. We also prove several corollaries of this identity, including a Santal type inequality and a contribution to the theory of summands. We proceed to discuss the analogous identity for convex bodies, where an unexpected distinction appears between the classical Minkowski addition and the more modern 2-addition. In the final section of the paper we consider the harmonic and geometric means of convex bodies and convex functions, and discuss their concavity properties. Once again, we find that in some problems the 2-addition of convex bodies behaves even better than the Minkowski addition.

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