Integral geometry

Source: Wikipedia, the free encyclopedia.

In

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 transforms such as the Radon transform
and its generalizations.

Classical context

Integral geometry as such first emerged as an attempt to refine certain statements of

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 of the plane acts. A 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 paradoxes—for example Bertrand's paradox.)

We can therefore say that integral geometry in this sense is the application of

Erlangen programme of Klein. The content of the theory is effectively that of invariant (smooth) measures on (preferably compact) homogeneous spaces of Lie groups; and the evaluation of integrals of the differential forms.[3]

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 processes concerned with geometric and incidence questions. See 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 theory.

The more recent meaning of integral geometry is that of

Sigurdur Helgason[4][5] and Israel Gelfand.[6] 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

  1. ^ Luis Santaló (1953) Introduction to Integral Geometry, Hermann (Paris)
  2. VEB Deutscher Verlag der Wissenschaften
  3. ISBN 0201135000
  4. ISBN 0821826735
  5. ISBN 9781441960542
  6. ISBN 0821829327

Further reading
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