A homogeneous decomposition theorem for valuations on convex functions

The existence of a homogeneous decomposition for continuous and epi-translation invariant valuations on super-coercive functions is established. Continuous and epi-translation invariant valuations that are epi-homogeneous of degree $n$ are classified. By duality, corresponding results are obtained for valuations on finite-valued convex functions.Comment: 16 page

The Lp-Minkowski problem for −n < p < 1

Chou and Wang's existence result for the Lp-Minkowski problem on Sn−1 for p∈(−n,1) and an absolutely continuous measure is discussed and extended to more general measures. In particular, we provide an almost optimal sufficient condition for the case p∈(0,1). © 2018 Elsevier Inc

Remarks on the KLS conjecture and Hardy-type inequalities

We generalize the classical Hardy and Faber-Krahn inequalities to arbitrary functions on a convex body $\Omega \subset \mathbb{R}^n$, not necessarily vanishing on the boundary $\partial \Omega$. This reduces the study of the Neumann Poincar\'e constant on $\Omega$ to that of the cone and Lebesgue measures on $\partial \Omega$; these may be bounded via the curvature of $\partial \Omega$. A second reduction is obtained to the class of harmonic functions on $\Omega$. We also study the relation between the Poincar\'e constant of a log-concave measure $\mu$ and its associated K. Ball body $K_\mu$. In particular, we obtain a simple proof of a conjecture of Kannan--Lov\'asz--Simonovits for unit-balls of $\ell^n_p$, originally due to Sodin and Lata{\l}a--Wojtaszczyk.Comment: 18 pages. Numbering of propositions, theorems, etc.. as appeared in final form in GAFA seminar note

Combination of Persistent Scatterer Interferometry and Single-Baseline Polarimetric Coherence Optimisation to Estimate Deformation Rates with Application to Tehran Basin

This study reports on the monitoring of land subsidence in a rural area located in the southwest of the Tehran basin, Iran, by combining a persistent scatterer interferometry (PSI) method with a single-baseline polarimetric coherence optimisation. Owing to vegetation coverage in this rural area, coherence level experiences a decline and the performance and coverage of conventional interferometry to estimate deformation rate reduces concomitantly. Since the launch of satellites with polarimetric information, the polarimetric InSAR (PolInSAR) technique, which is vector interferometry with different polarimetric channels, has been introduced to optimise the coherence level. One of the most common criteria to select PS pixels is coherence and maximising the coherence can lead to an increased number of selected PS pixels and enhanced PSI performance. The single-baseline polarimetric coherence optimisation method assumes equal polarisation states at the end of each baseline. In order to apply this technique in our study, two different multi-look windows for coherence calculation and also two TerraSAR-X data sets with different numbers of images are used to assess their effect on the polarimetric PSI. Combination of the single-baseline coherence optimisation method with PSI shows significant improvements (more than 50%) in terms of the number of selected PS pixels in the case study even using a data set with a small number of images. A 15 × 15 multi-look window selects a greater number of PS pixels compared to a 9×9 multi-look window, although this entails reducing spatial resolution. The most effective PSI approach in terms of the density of the selected PS turned out to be polarimetric PSI using a data set with a large number of images and a selection of a 15 × 15 multi-look window. Validation of the PSI methods using a large number of images with 9×9 and 15 × 15 multi-look windows via levelling measurements confirms the accuracy and reliability of the results obtained

Isoperimetry and stability of hyperplanes for product probability measures

International audienceWe investigate stationarity and stability of half-spaces as isoperimetric sets for product probability measures, considering the cases of coordinate and non-coordinate half-spaces. Moreover, we present several examples to which our results can be applied, with a particular emphasis on the logistic measure