# 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} }

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.

## 16 Citations

### Curvature functionals on convex bodies

- MathematicsCanadian Mathematical Bulletin
- 2022

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

- Mathematics, Computer Science
- 2014

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.

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

- MathematicsIndiana University Mathematics Journal
- 2022

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

- MathematicsCanadian Mathematical Bulletin
- 2017

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

- MathematicsThe Journal of Geometric Analysis
- 2022

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

- MathematicsStatistics Surveys
- 2019

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

- Mathematics, Philosophy
- 2014

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

- Mathematics
- 2014

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

- Mathematics
- 2012

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…

### Inequalities for mixed p-affine surface area

- Mathematics
- 2010

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

- Mathematics
- 1997

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

- Mathematics
- 2011

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

- Mathematics
- 1999

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…

### Sharp Affine LP Sobolev Inequalities

- Mathematics
- 2002

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

- Mathematics
- 2014

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…