Isoperimetric Inequalities, Brunn–Minkowski Theory and Minkowski-Type Monge–Ampère Equations on the Sphere

Károly J. Böröczky
Alfréd Rényi Institute of Mathematics, Budapest, Hungary
Alessio Figalli
ETH Zürich, Switzerland
João P. G. Ramos
Instituto Nacional de Matemática Pura e Aplicada, Rio de Janeiro, Brazil

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	Front matterDownload pp. i–iv
	PrefaceDownload pp. vii–ix
	ContentsDownload pp. xi–xv
	Basic notions and notationpp. 1–4
1	Preliminaries in convex geometry in $R^{n}$ pp. 5–60
2	Surface area, surface area measure and cone volume measure for convex bodies in $R^{n}$ pp. 61–92
3	The Brunn–Minkowski and the Prékopa–Leindler inequalities in the measurable casepp. 93–120
4	The isoperimetric inequality in the case of Lipschitz boundarypp. 121–176
5	The isoperimetric inequality for sets of finite perimeter in $R^{n}$ and the Sobolev inequality for BV functionspp. 177–202
6	Associated ellipsoids, Blaschke–Santaló inequality and the reverse isoperimetric inequalitypp. 203–260
7	Steiner formula and mixed volumespp. 261–302
8	Convex bodies and Gaussian curvaturepp. 303–390
9	The Minkowski problem, the $L_{p}$ -Minkowski problem, and the $L_{p}$ -Brunn–Minkowski inequality/conjecturepp. 391–454
A	Appendix: Background from analysis and algebrapp. 455–480
	Referencespp. 481–519
	Indexpp. 521–523