Geodesic
An application of integral geometry to linear inequality theory
Zapiski Nauchnykh Seminarov POMI, Differential geometry, Lie groups and mechanics. Part V, Tome 123 (1983), pp. 208-220

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A. M. Versik suggested to identify Linear Programming Problems space with the corresponding Grassmann manifold. A probablility measure is defined on the manifold. The average number of permissible bases and the measures of the problems with finite and infinite extreme are calculated. 
@article{ZNSL_1983_123_a16,
     author = {P. V. Sporyshev},
     title = {An application of integral geometry to linear inequality theory},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {208--220},
     publisher = {mathdoc},
     volume = {123},
     year = {1983},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1983_123_a16/}
}
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P. V. Sporyshev. An application of integral geometry to linear inequality theory. Zapiski Nauchnykh Seminarov POMI, Differential geometry, Lie groups and mechanics. Part V, Tome 123 (1983), pp. 208-220. http://geodesic.mathdoc.fr/item/ZNSL_1983_123_a16/