Operations between functions

Gardner, Richard J.; Kiderlen, Markus

doi:10.4310/cag.2018.v26.n4.a5

Cited by 6 publications

(14 citation statements)

References 25 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…if φ is concave; the direction of the inequality is reversed if φ is convex. If φ is strictly concave or convex and (20) is equivalent to inequality (21) in the following sense: if one of them holds, the other one also holds.…”

Section: An Inequality Equivalent To Jensen's Inequalitymentioning

confidence: 99%

“…Although in Definition 1, the functions p 1 , · · · , p m are assumed to be measurable, equation (2) can also be used to define the Orlicz addition of general nonnegative functions. In an independent work [21], Gardner and Kiderlen also provided the definition for the Orlicz addition of nonnegative functions. The second author of this paper would like to thank Professor Gardner for mentioning [21] to him.…”

Section: Introductionmentioning

confidence: 99%

“…interpretation of the f -divergence and related inequalities; while the paper [21] mainly aims to provide a structural theory of operations between real-valued functions.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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

Hou

2019

Can. J. Math.-J. Can. Math.

View full text Add to dashboard Cite

In this paper, the Orlicz addition of measures is proposed and an interpretation of the f -divergence is provided based on a linear Orlicz addition of two measures. Fundamental inequalities, such as, a dual functional Orlicz-Brunn-Minkowski inequality, are established. We also investigate an optimization problem for the f -divergence and establish functional affine isoperimetric inequalities for the dual functional Orlicz affine and geominimal surface areas of measures. Index Termsaffine isoperimetric inequality, affine surface area, Brunn-Minkowski theory, dual Brunn-Minkowski theory, the f -divergence, geominimal surface area, optimization problem for the f -divergence.with equality if and only if p = q almost everywhere with respect to Q. When f is strictly concave with f (1) = 0, one gets similar results with "≥" replaced by "≤". From this viewpoint, the f -divergence can be used to distinguish two measures. Without doubt, the f -divergence plays fundamental roles in, such as, image analysis, information September 11, 2018 DRAFT 2 theory, pattern matching and statistical learning (see [5], [11], [24], [27], [34]), where the measure of difference between measures is required. Moreover, in general, the f -divergence is arguably better than the L p distance. Recent development in convex geometry has witnessed the strong connections between the f -divergence and convex geometry. For instance, it has been proved that the L p affine surface area [7], [32], [37], a central notion in convex geometry, is related to the Renyi entropy [38]; while the general affine surface area [28], [30] is associated to the f -divergence [9]. Note that these affine surface areas are valuations; and valuations are the key ingredients for the Dehn's solution of Hilbert third problem. Moreover, under certain conditions (such as, semicontinuity), it has been proved that these affine surface areas can be used to uniquely characterize all valuations which remain unchanged under linear transforms with determinant ±1 (see e.g. [23], [29], [30]). On the other hand, as showed in the Subsection V-A, the affine and geominimal surface areas (see e.g. [32], [35], [40], [41]) can be translated to an optimization problem for the f -divergence. This observation leads us to investigate the dual functional Orlicz affine and geominimal surface areas for measures, which are invariant under linear transforms with determinant ±1. The Brunn-Minkowsi inequality is arguably one of the most important inequalities in convex geometry. It can be used to prove, for instance, the celebrated Minkowski's and isoperimetric inequalities. (Note that the isoperimetric problem has a history over 1000 years). See the excellent survey [18] for more details. On the other hand, the dual Brunn-Minkowski inequality and dual Minkowski inequality are crucial for the solutions of the famous Busemann-Petty problem (see e.g., [17], [22], [31], [43]). The Brunn-Minkowsi inequality and its dual have been extended to the Orlicz theory in [19], [20], [39], [44].This paper is dedicated to...

show abstract

Section: An Inequality Equivalent To Jensen's Inequalitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

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

Hou

2019

Can. J. Math.-J. Can. Math.

View full text Add to dashboard Cite

show abstract

“…, and in each case D 2 is the common domain of F + and F − . The pointwise property is inspired by the pointwise operations defined in [21]. The functions F + and F − are said to be associated with T .…”

Section: (Maps Balls To Balls)mentioning

confidence: 99%

“…Again taking a cue from [21], one might consider the following more general version of the pointwise property:…”

Section: (Maps Balls To Balls)mentioning

confidence: 99%

Rearrangement and polarization

Bianchi,

Gardner,

Gronchi

et al. 2019

Preprint

Self Cite

View full text Add to dashboard Cite

The paper has two main goals. The first is to take a new approach to rearrangements on certain classes of measurable real-valued functions on R n . Rearrangements are maps that are monotonic (up to sets of measure zero) and equimeasurable, i.e., they preserve the measure of super-level sets of functions. All the principal known symmetrization processes for functions, such as Steiner and Schwarz symmetrization, are rearrangements, and these have a multitude of applications in diverse areas of the mathematical sciences. The second goal is to understand which properties of rearrangements characterize polarization, a special rearrangement that has proved particularly useful in a number of contexts. In order to achieve this, new results are obtained on the structure of measure-preserving maps on convex bodies and of rearrangements generally.

show abstract