The classical subjects of geometric probability and integral geometry, and the more modern one of stochastic geometry, are developed here in a novel way to provide a framework in which they can be studied. Integral geometry and geometric probability luis antonio. Now available in the cambridge mathematical library, the classic work from luis santali. Cambridge core geometry and topology integral geometry and geometric probability by luis a. Among the several terms used to describe such investigations are stochastic or statistical geometry, integral geometry, and geometric probability. This book originated from two courses delivered by the authors at the crm and provides a selfcontained introduction to. First book since santalos classic 1976 book to combine stochastic geometry and integral geometry. Absolute curvatures in integral geometry mathematical.
It rigorously presents foundations of the models of stochastic geometry and of the tools from integral geometry, and it supplies clear, complete, and. In the last years there has been significant progress in the theory of valuations, which in turn has led to important achievements in integral geometry. This formula is due to blaschke, santalo and chern. The integral geometric approach to geometrical quantities harvard. Contributions to science, institut destudis catalans. Between the 1950s and the 1970s, he wrote a series of papers on general relativity and on the. Pdf stochastic and integral geometry probability and its. Radon in 1917, who obtained an explicit formula recovering a compactly supported c 8 function on r3 from its plane integrals. Integral geometry and geometric probability cambridge mathematical library by luis a. Santalo, integral ge ometry and ge ometric prob ability. Kaiser department of industrial engineering auburn university, auburn, al 36849, u.
The abstract phase of the last decade is illustrated by some work of the author. Integral geometry and valuations semyon alesker springer. The santalo point of a function, and a functional form of. Buy integral geometry and geometric probability cambridge mathematical library on. Calculus done indirectlyby local mapping to proofs requiredto establish results independent of coordinates. Work in integral geometry dates back at least to the work of crofton, who showed that the invariant measure of the set of lines intersecting an arc in the plane is proportional to the arclength.
Some integral geometry geometric probability derive the santalo proof of the isoperimetric and bonnesen inequalities goal. It is a standard tool in integral geometry and has applications in isoperimetric and rigidity results. He was an outstanding exponent of integral geometry, a great teacher and disseminator of scientific understanding, and a person of great hu. The short survey paper of viro 78 cited macpherson and mentioned simple applications of the euler integral to algebraic geometry. A subriemannian santalo formula with applications to isoperimetric inequalities and first dirichlet eigenvalue of hypoelliptic operators prandi, dario, rizzi, luca, and seri, marcello, journal of differential geometry, 2019. The santalo point of a planar convex set sciencedirect. A combinatorial problem in geometry 1935b alexandroff and hopf, topologie 1936a santalo, integral geometry 2 1936b wagner any triangulation can be transformed into any other by flips 1948 fary straight line embeddings of graphs 1951 tarski. Santalo, integral geometry and geometric probability, encyclopedia of mathematics and its applications, vol. Its later developments have proved to be useful in several fields ranging from pure mathematics measure theory, continuous groups to technical and applied disciplines pattern recognition, stereology. Read integral geometry and geometric probability cambridge mathematical library by luis a.
Santalo, integral geometry on surfaces of constant negative curvature, duke. Integral geometry and geometric inequalities in hyperbolic space article in differential geometry and its applications 223. Between the 1950s and the 1970s, he wrote a series of papers on general relativity. Santalo, integral geometry and geometric probability, addisonwesley, 1976. Notes on the integral geometry in the hyperbolic plane core.
Introduction the study of probabilities on geometric objects has encompassed a wide variety of approaches and concerns. Santalo, integral geometry and geometric probability chern, s. Its later developments, however, have proved to be useful in several fields ranging from pure mathematics measure theory, continuous groups to technical and applied disciplines pattern recognition. The integral geometry in the hyperbolic plane was initiated many years ago in the article integral geometry on surfaces of constant negative curvature, written by ll. Ball about even functions, and prove that for any measurable function f. Integral geometry and geometric probability cambridge. The book is a systematic exposition of the theory and a compilation of. This strand of integral geometry goes back to the work of p. Between the 1950s and the 1970s, he wrote a series of papers on general. Cambridge university press 9780521523448 integral geometry and geometric probability, second.
Its later developments, however, have proved to be useful in several fields ranging from pure mathematics measure theory, continuous groups to technical and. Integral geometry originated with problems on geometrical probability and convex bodies. Of great significance in integral geometry are measurable lie groups, that is, groups that admit one and only one invariant up to a constant factor. Funk 4 in 1916, who showed that a continuous even function on the twosphere can be recovered from its integrals over great circles, as well as that of j. Its later developments, however, have proved to be useful in several fields ranging from pure mathematics measure theory, continuous groups to technical and applied disciplines pattern recognition, stereology. The book is a systematic exposition of the theory and a compilation of the main results in the field. This book provides the readers with new findings, some being published for the first time and serves as an excellent graduate text. Eulercalculus withapplications tosignals and sensing. Kinematic formula, dual quermassintegral, chord power integral, dual. Surface integrals of curvature arise naturally in integral geometry and geometrical probability, most often in connection with the quermassintegrale or crosssection integrals of convex bodies. Luis santalo and classical field theory springerlink. Integral geometry from buffon to geometers of today. Stochastic and integral geometry rolf schneider springer. Integral geometry and geometric probability edition 2 by.
Kinematic and crofton formulae of integral geometry. The language of euler calculus was slow to form and be appreciated. Integral geometry and geometric probability by luis a. His work has applications in number theory, in the theory of differential equations, in stochastic geometry, in functional analysis, and also in theoretical physics. The book is a systematic exposition of the theory and. Integral geometry originated from geometric probability. Demidova, application of the theory of integral invariants of lie groups to certain aspects of integral geometry, dissertation, kharkov 1956. Known results and open problems in hypercomplex convexity. The image is appropriate, for the subject of integral geometry originated in buffons needle problem. Integral geometry and geometric inequalities in hyperbolic. Integral geometry studies the link between expectation of random variables and. In the late 18th century, buffon proved that the probability that a random line. Integral geometry and geometric probability bookask. Essentials of integral geometry in a homogenous space are presented and the focus is on the basic results and applications.
Abstract let lf denotes the legendre transform of a function f. They enjoy many desirable properties, such as the ability to be determined by summing or averaging over lowerdimensional sections or projections. Santalo, integral geometry and geometric probability. Topics in integral geometry series in pure mathematics.