View source for Integral geometry

In [[Mathematics|mathematics]], '''integral geometry''' is the theory of [[Measure (mathematics)|measures]] on a geometrical space invariant under the [[Symmetry group|symmetry group]] of that space.  In more recent times, the meaning has been broadened to include a view of invariant (or equivariant) transformations from the space of functions on one geometrical space to the space of functions on another geometrical space.  Such transformations often take the form of [[Integral transform|integral transform]]s such as the [[Radon transform]] and its generalizations.
== Classical context ==
Integral geometry as such first emerged as an attempt to refine certain statements of [[Geometric probability|geometric probability theory]].  The early work of [[Biography:Luis Santaló|Luis Santaló]]<ref>Luis Santaló (1953) ''Introduction to Integral Geometry'', Hermann (Paris)</ref> and [[Biography:Wilhelm Blaschke|Wilhelm Blaschke]]<ref>Wilhelm Blaschke (1955) ''Vorlesungen über Integralgeometrie'',  VEB Deutscher Verlag der Wissenschaften</ref> was in this connection. It follows from the [[Crofton formula|classic theorem of Crofton]] expressing the [[Physics:Length|length]] of a plane [[Curve|curve]] as an [[Expected value|expectation]] of the number of intersections with a random line.  Here the word 'random' must be interpreted as subject to correct symmetry considerations.

There is a sample space of lines, one on which the [[Affine group|affine group]] of the plane acts. A [[Probability measure|probability measure]] is sought on this space, invariant under the symmetry group. If, as in this case, we can find a unique such invariant measure, then that solves the problem of formulating accurately what 'random line' means and expectations become integrals with respect to that measure. (Note for example that the phrase 'random chord of a circle' can be used to construct some [[Philosophy:Paradox|paradox]]es—for example [[Bertrand paradox (probability)|Bertrand's paradox]].)

We can therefore say that integral geometry in this sense is the application of [[Probability theory|probability theory]] (as axiomatized by Kolmogorov) in the context of the Erlangen programme of [[Biography:Felix Klein|Klein]]. The content of the theory is effectively that of invariant (smooth) measures on (preferably [[Compact space|compact]]) [[Homogeneous space|homogeneous space]]s of [[Lie group]]s; and the evaluation of integrals of the [[Differential form|differential form]]s.<ref>Luis Santaló (1976) ''Integral Geometry and Geometric Probability'', Addison Wesley {{ISBN|0201135000}}</ref>

A very celebrated case is the problem of [[Buffon's needle]]: drop a needle on a floor made of planks and calculate the probability the needle lies across a crack. Generalising, this theory is applied to various [[Stochastic process|stochastic process]]es concerned with geometric and incidence questions.  See [[Stochastic geometry|stochastic geometry]].

One of the most interesting theorems in this form of integral geometry is [[Hadwiger's theorem]] in the Euclidean setting. Subsequently Hadwiger-type theorems were established in various settings, notably in hermitian geometry, using advanced tools from [[Valuation (geometry)|valuation theory]].

The more recent meaning of '''integral geometry''' is that of Sigurdur Helgason<ref>Sigurdur Helgason (2000) ''Groups and Geometric Analysis: integral geometry, invariant differential operators, and spherical functions'', American Mathematical Society {{ISBN|0821826735}}</ref><ref>Sigurdur Helgason (2011) ''Integral Geometry and Radon Transforms'', Springer, {{ISBN|9781441960542}}
</ref> and [[Biography:Israel Gelfand|Israel Gelfand]].<ref>I.M. Gel’fand (2003) ''Selected Topics in Integral Geometry'', American Mathematical Society {{ISBN|0821829327}}</ref> It deals more specifically with integral transforms, modeled on the [[Radon transform]]. Here the underlying geometrical incidence relation (points lying on lines, in Crofton's case) is seen in a freer light, as the site for an integral transform composed as ''pullback onto the incidence graph'' and then ''push forward''.

==Notes==
{{Reflist}}

==Further reading==
*Sors, Luis Antonio Santaló, and Luis A. Santaló. ''Integral geometry and geometric probability''. Cambridge university press, 2004. A systematic exposition of the theory and a compilation of the main results.
*Langevin, Rémi. Integral geometry from Buffon to geometers of today. Vol. 23. SMF, 2016. A more elementary exposition, focusing on the [[Crofton formula]] and generalizations thereof.
*{{springer|title=Integral geometry|id=I/i051470|first=S.F|last=Shushurin|year=2001}}


[[Category:Integral geometry| ]]

{{Sourceattribution|Integral geometry}}