Convex bodies: the brunn–minkowski theory

Introduction What are Poincare and Sobolev inequalities? Poincare inequalities, pointwise estimates, and Sobolev classes Examples and necessary conditions Sobolev type inequalities by means of Riesz potentials Trudinger inequality A version of the Sobolev embedding theorem on spheres Rellich-Kondrachov Sobolev classes in John domains Poincare inequality: examples Carnot-Caratheodory spaces Graphs Applications to PDE and nonlinear potential theory Appendix References.

Sobolev met Poincaré

After Hormander's fundamental paper on hypoellipticity [54], the study of partial differential equations arising from families of noncommuting vector fields has developed significantly. In this paper we study some basic functional and geometric properties of general families of vector fields that include the Hormander type as a special case. Similar to their classical counterparts, such properties play an important role in the analysis of the relevant differential operators (both linear and nonlinear). To motivate our results, we recall some classical inequalities. Let E C R" be a Caccioppoli set (a measurable set having a locally finite perimeter); then one has the isoperimetric inequality

Isoperimetric and Sobolev inequalities for Carnot-Carathéodory spaces and the existence of minimal surfaces

We present a mathematical model of perceptual completion and formation of subjective surfaces, which is at the same time inspired by the architecture of the visual cortex, and is the lifting in the 3-dimensional rototranslation group of the phenomenological variational models based on elastica functional. The initial image is lifted by the simple cells to a surface in the rototranslation group and the completion process is modeled via a diffusion driven motion by curvature. The convergence of the motion to a minimal surface is proved. Results are presented both for modal and amodal completion in classic Kanizsa images.

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A Cortical Based Model of Perceptual Completion in the Roto-Translation Space

Geometry of sets and measures in euclidean spaces fractals and rectifiability

The Isoperimetric Problem in Euclidean Space.- The Heisenberg Group and Sub-Riemannian Geometry.- Applications of Heisenberg Geometry.- Horizontal Geometry of Submanifolds.- Sobolev and BV Spaces.- Geometric Measure Theory and Geometric Function Theory.- The Isoperimetric Inequality in ?.- The Isoperimetric Profile of ?.- Best Constants for Other Geometric Inequalities on the Heisenberg Group.

An Introduction to the Heisenberg Group and the Sub-Riemannian Isoperimetric Problem

(1993). An embedding theorem and the harnack inequality for nonlinear subelliptic equations. Communications in Partial Differential Equations: Vol. 18, No. 9-10, pp. 1765-1794.

An embedding theorem and the harnack inequality for nonlinear subelliptic equations

http://www.intlpress.com/site/pub/files/_fulltext/journals/cag/1994/0002/0002/CAG-1994-0002-0002-a002.pdf

The geometric Sobolev embedding for vector fields and the isoperimetric inequality

CONTENTS 1. Introduction 2. Preliminaries 3. Strongly convex functions: A notion to abandon 4. Mildly convex functions: The appearance of horizontal planes 5. Weakly convex functions: The horizontal Hessian 6. Weak convexity of the gauge in groups of Heisenberg type 7. Convexity of sets 8. Further properties of second derivatives of weakly convex functions 9. Basic properties of first derivatives of weakly convex functions: L

https://www.intlpress.com/site/pub/files/_fulltext/journals/cag/2003/0011/0002/CAG-2003-0011-0002-a005.pdf

Donatella Danielli

Papers

An Introduction to the Heisenberg Group and the Sub-Riemannian Isoperimetric Problem

An embedding theorem and the harnack inequality for nonlinear subelliptic equations

The geometric Sobolev embedding for vector fields and the isoperimetric inequality

Notions of Convexity in Carnot Groups

Sub-Riemannian calculus on hypersurfaces in Carnot groups