f-Divergence for convex bodies

@article{Werner2012fDivergenceFC,
  title={f-Divergence for convex bodies},
  author={Elisabeth M. Werner},
  journal={ArXiv},
  year={2012},
  volume={abs/1205.3423}
}
  • E. Werner
  • Published 15 May 2012
  • Mathematics
  • ArXiv
We introduce f-divergence, a concept from information theory and statistics, for convex bodies in ℝ n . We prove that f-divergences are SL(n) invariant valuations and we establish an affine isoperimetric inequality for these quantities. We show that generalized affine surface area and in particular the L p affine surface area from the L p Brunn Minkowski theory are special cases of f-divergences. 

Curvature functionals on convex bodies

We investigate the weighted L p affine surface areas which appear in the recently established L p Steiner formula of the L p Brunn Minkowski theory. We show that they are valuations on the set of

Mixed f‐divergence and inequalities for log‐concave functions

Finite invariant vector entropy inequalities, like new Alexandrov–Fenchel‐type inequalities and an affine isoperimetric inequality for the vector form of the Kullback Leibler divergence for log‐concave functions, are introduced.

Divergence for s-concave and log concave functions

Pinsker inequalities and related Monge-Ampere equations for log-concave functions

In this paper we further develop the theory of f-divergences for log-concave functions and their related inequalities. We establish Pinsker inequalities and new affine invariant entropy inequalities.

Mixed ƒ-divergence for Multiple Pairs of Measures

An Alexandrov–Fenchel type inequality and an isoperimetric inequality for the mixed $f -divergence are proved and properties for the mix of measures are established, such as permutation invariance and symmetry in distributions.

Affine Invariant Maps for Log-Concave Functions

Affine invariant points and maps for sets were introduced by Grünbaum to study the symmetry structure of convex sets. We extend these notions to a functional setting. The role of symmetry of the set

Halfspace depth and floating body

Little known relations of the renown concept of the halfspace depth for multivariate data with notions from convex and affine geometry are discussed. Halfspace depth may be regarded as a measure of

Lp Geominimal Surface Areas and their Inequalities

In this paper, we introduce the Lp geominimal surface area for all −n 6 p 1). Our extension of the Lp geominimal surface area is motivated by recent work on the extension of the Lp affine surface

On the monotone properties of general affine surfaces under the Steiner symmetrization

In this paper, we prove that, if functions (concave) $\phi$ and (convex) $\psi$ satisfy certain conditions, the $L_{\phi}$ affine surface area is monotone increasing, while the $L_{\psi}$ affine

References

SHOWING 1-10 OF 58 REFERENCES

Relative entropy of cone measures and Lp centroid bodies

Let K be a convex body in ℝn. We introduce a new affine invariant, which we call ΩK, that can be found in three different ways: as a limit of normalized Lp‐affine surface areas; as the relative

R\'enyi Divergence and $L_p$-affine surface area for convex bodies

Inequalities for mixed p-affine surface area

We prove new Alexandrov-Fenchel type inequalities and new affine isoperimetric inequalities for mixed p-affine surface areas. We introduce a new class of bodies, the illumination surface bodies, and

On the p-Affine Surface Area

We give geometric interpretations of certain affine invariants of convex bodies. The affine invariants are the p-affine surface areas introduced by Lutwak. The geometric interpretations involve

Relative entropies for convex bodies

We introduce a new class of (not necessarily convex) bodies and show, among other things, that these bodies provide yet another link between convex geometric analysis and information theory. Namely,

Convolutions, Transforms, and Convex Bodies

The paper studies convex bodies and star bodies in Rn by using Radon transforms on Grassmann manifolds, p‐cosine transforms on the unit sphere, and convolutions on the rotation group of Rn. It

Intersection bodies and dual mixed volumes

Sharp Affine LP Sobolev Inequalities

A sharp affine Lp Sobolev inequality for functions on Euclidean n-space is established. This new inequality is significantly stronger than (and directly implies) the classical sharp Lp Sobolev

The Discrete Planar L 0-Minkowski Problem

In the discrete setting, the L0-Minkowski problem extends the question posed and answered by the classical Minkowski’s existence theorem for polytopes. In particular, the planar extension, which we
...