Convex sets and convex functions: the fundamentals

2006 ◽  
pp. 1-19 ◽  
Author(s):  
Roberto Lucchetti
Keyword(s):  
Convex Functions ◽  
Convex Sets

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.

scholarly journals Valuations on convex functions and convex sets and Monge–Ampère operators

2019 ◽  
Vol 19 (3) ◽  
pp. 313-322 ◽  
Author(s):  
Semyon Alesker
Keyword(s):  
Convex Functions ◽  
Convex Sets ◽  
Convex Bodies ◽  
Linear Functionals ◽  
Translation Invariant ◽  
Infinite Dimensional ◽  
Dense Image ◽  
Monge Ampere ◽  
Explicit Relation ◽  
Class Of Functions

Abstract The notion of a valuation on convex bodies is very classical; valuations on a class of functions have been introduced and studied by M. Ludwig and others. We study an explicit relation between continuous valuations on convex functions which are invariant under adding arbitrary linear functionals, and translation invariant continuous valuations on convex bodies. More precisely, we construct a natural linear map from the former space to the latter and prove that it has dense image and infinite-dimensional kernel. The proof uses the author’s irreducibility theorem and properties of the real Monge–Ampère operators due to A.D. Alexandrov and Z. Blocki. Furthermore we show how to use complex, quaternionic, and octonionic Monge–Ampère operators to construct more examples of continuous valuations on convex functions in an analogous way.

scholarly journals Well-positioned Closed Convex Sets and Well-positioned Closed Convex Functions

2004 ◽  
Vol 29 (4) ◽  
pp. 337-351 ◽  
Author(s):  
Samir Adly ◽  
Emil Ernst ◽  
Michel Théra
Keyword(s):  
Convex Functions ◽  
Convex Sets
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