### Extremal properties of Singe's world function and discrete geometry

#### Yu.A. Rylov

Institute for Problems in Mechanics, Academy of Sciences of the USSR,
Prospect Vernadskogo 101-1 , Moscow 117526, USSR

email: rylov@ipmnet.ru

Properties of sigma space
[a set of points *P *with real function sigma(*P,P')* given on this
set] are investigated. A continuity of the set is not necessary and, generally,
geometry is discrete. The properties of the world function sigma are
investigated. At certain (extremal) world function properties the sigma space
is shown to be a subset of points of Euclidean space or Riemannian space. The
presented approach has the pecularity that no operation other than the world
function is given on sigma space. In particular, all such operations as linear
operation over vectors, constructing lines and planes, and dimension of the
space are expressed through the world function and only through it (if it is
extremal). A violation of the sigma-space extremality leads to going out beyond
the frames of Riemannian geometry (lines are substituted by tubes of lines,
etc.). The presented approach can be useful in quantum gravitation, string
models, and other problems, where the properties of the event space at small
distances are important.

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