Between the 1950s and the 1970s, he wrote a series of papers on general relativity. 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. Santalo, integral geometry and geometric probability chern, s. Santalo, integral geometry on surfaces of constant negative curvature, duke. Integral geometry and geometric inequalities in hyperbolic. Integral geometry studies the link between expectation of random variables and. Integral geometry and geometric probability cambridge mathematical library by luis a. This strand of integral geometry goes back to the work of p. 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. The abstract phase of the last decade is illustrated by some work of the author. Kinematic formula, dual quermassintegral, chord power integral, dual. It is a standard tool in integral geometry and has applications in isoperimetric and rigidity results. Its later developments, however, have proved to be useful in several fields ranging from pure mathematics measure theory, continuous groups to technical 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.
The santalo point of a function, and a functional form of. His work has applications in number theory, in the theory of differential equations, in stochastic geometry, in functional analysis, and also in theoretical physics. Integral geometry originated with problems on geometrical probability and convex bodies. Cambridge university press 9780521523448 integral geometry and geometric probability, second. Santalo, integral geometry and geometric probability, addisonwesley, 1976. Absolute curvatures in integral geometry mathematical. Integral geometry and geometric inequalities in hyperbolic space article in differential geometry and its applications 223. 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. This book originated from two courses delivered by the authors at the crm and provides a selfcontained introduction to. Known results and open problems in hypercomplex convexity. The book is a systematic exposition of the theory and a compilation of the main results in the field. The book is a systematic exposition of the theory and a compilation of.
Cambridge core geometry and topology integral geometry and geometric probability by luis a. Eulercalculus withapplications tosignals and sensing. 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 image is appropriate, for the subject of integral geometry originated in buffons needle problem. He was recognized as one of the great geometers of the century integral geometry and geometric probability 1976, but also made original contributions to unified field theory and studies on the history of mathematics. They enjoy many desirable properties, such as the ability to be determined by summing or averaging over lowerdimensional sections or projections.
This formula is due to blaschke, santalo and chern. The santalo point of a planar convex set sciencedirect. Contributions to science, institut destudis catalans. Between the 1950s and the 1970s, he wrote a series of papers on general relativity and on the. The short survey paper of viro 78 cited macpherson and mentioned simple applications of the euler integral to algebraic geometry. Stochastic and integral geometry rolf schneider springer. This book provides the readers with new findings, some being published for the first time and serves as an excellent graduate text. Integral geometry and geometric probability cambridge.
Now available in the cambridge mathematical library, the classic work from luis santali. Essentials of integral geometry in a homogenous space are presented and the focus is on the basic results and applications. Among the several terms used to describe such investigations are stochastic or statistical geometry, integral geometry, and geometric probability. Between the 1950s and the 1970s, he wrote a series of papers on general. Abstract let lf denotes the legendre transform of a function f. Integral geometry from buffon to geometers of today. Considered one of the founding fathers of integral geometry, luis santalo has contributed to various areas of mathematics. Integral geometry and geometric probability pdf free download. Integral geometry originated from geometric probability. Some integral geometry geometric probability derive the santalo proof of the isoperimetric and bonnesen inequalities goal. Demidova, application of the theory of integral invariants of lie groups to certain aspects of integral geometry, dissertation, kharkov 1956. Kaiser department of industrial engineering auburn university, auburn, al 36849, u. Introduction the study of probabilities on geometric objects has encompassed a wide variety of approaches and concerns.
He was an outstanding exponent of integral geometry, a great teacher and disseminator of scientific understanding, and a person of great hu. 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. In the late 18th century, buffon proved that the probability that a random line. Integral geometry and geometric probability edition 2 by.
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. Calculus done indirectlyby local mapping to proofs requiredto establish results independent of coordinates. Integral geometry and valuations semyon alesker springer. Buy integral geometry and geometric probability cambridge mathematical library on. 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.
It rigorously presents foundations of the models of stochastic geometry and of the tools from integral geometry, and it supplies clear, complete, and. Integral geometry and geometric probability by luis a. Notes on the integral geometry in the hyperbolic plane core. Santalo, integral geometry and geometric probability. Topics in integral geometry series in pure mathematics. Radon in 1917, who obtained an explicit formula recovering a compactly supported c 8 function on r3 from its plane integrals. 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. Read integral geometry and geometric probability cambridge mathematical library by luis a. Santalo, integral ge ometry and ge ometric prob ability. 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. 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.
The integral geometric approach to geometrical quantities harvard. Ball about even functions, and prove that for any measurable function f. First book since santalos classic 1976 book to combine stochastic geometry and integral geometry. The book is a systematic exposition of the theory and. Kinematic and crofton formulae of integral geometry. The language of euler calculus was slow to form and be appreciated. Luis santalo and classical field theory springerlink. Integral geometry and geometric probability bookask. 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. Santalo, integral geometry and geometric probability, encyclopedia of mathematics and its applications, vol. Pdf stochastic and integral geometry probability and its.
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