R\'enyi Divergence and $L_p$-affine surface area for convex bodies
By Elisabeth M. Werner. 2011
94 citations recorded by the source corpus.
We show that the fundamental objects of the $L_p$-Brunn-Minkowski theory, namely the $L_p$-affine surface areas for a convex body, are closely related to information theory: they are exponentials of R\'enyi divergences of the cone measures of a convex body and its polar. We give geometric interpretations for all R\'enyi divergences $D_\alpha$, not just for the previously treated special case of relative entropy which is the case $\alpha =1$. Now, no symmetry assumptions are needed and, if at all, only very weak regularity assumptions are required. Previously, the relative entropies appeared only after performing second order expansions of certain expressions. Now already first order expansions makes them appear. Thus, in the new approach we detect "faster" details about the boundary of a convex body.
Subfields: Applied Mathematics
View Paper: https://www.semanticscholar.org/paper/2e46cb80ba0a5f5cef8cf3f353edfc57e5c393c5
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
In Applied Mathematics
Citations in the corpus
- Dual Orlicz–Brunn–Minkowski theory: Dual Orlicz Lϕ affine and geominimal surface areas (2014, background): “These affine invariants are very useful in applications, see [15, 17, 21, 22, 23, 35, 40, 43, 44] among others.”
- THE ORLICZ-BRUNN-MINKOWSKI THEORY: A GENERAL FRAMEWORK, ADDITIONS, AND INEQUALITIES (2013, unknown): “See, for example, [4], [13], [14], [21], [22], [23], [24], [26], and [35].”