Integral geometry

In Mathematical, the term integral geometry is used to indicate a vast whole of results on the geometrical interpretation of certain results of Integral calculus.

Fields

The traditional use of this term is that introduced by Santalo and Blaschke. It comes from the Formule of Crofton making it possible to express the Length of a plane Courbe like the mathematical Espérance of the number of intersection with a line of the plan Aléatoire. Here, the random term is to be taken according to considerations of symmetries.

There exists a space of lines of the plan, on which the Groupe refines plan acts. A measurement of probability can provide this space, invariant by the group of symmetries. If, as in this case, one finds only one single measurement invariant, that résoud the problem to precisely formulate what the random term right wants to say; the hope mathematics thus amounts carrying out an integration with this measurement.

One can then see the integral geometry within the meaning of Santalo , like the application of the Theory of probability (as axiomatized by Kolmogorov) in the context of the Programme of Erlangen of Klein. Indeed, this approach consists of the use of invariant measurement on homogeneous spaces (preferably compact) of groups of Dregs; and the evaluation of integrals on differential forms.

A very famous application is the problem of the Aiguille of Buffon: to make fall a needle on a parquet floor from slat then to calculate the probability that this one is on at least 2 slats. More generally, this approach can be applied to many stochastic Processus S in relation to geometrical problems.

One of the most famous résulats of this field is the Théorème of Hadwiger.

The most recent intrepretation of the term of integral geometry is that of Israel Gelfand. It applies to the integral transformations, utilizing the transformed Radon.

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