Faint Structures in Low Density Regions of the Nearby Universe

We present an investigation of the galaxy distribution in the huge underdense region between the Hercules, Coma and Local Superclusters, the so-called Northern Local Void (NLV), using void statistics (for details refer to Lindner et al. this Volume). Reshift data for galaxies and poor clusters of galaxies are available in low and high density regions as well. Samples of galaxies with different morphological type and various luminosity limits have been studied separately and void catalogues have been compiled from three different luminosity limited galaxy samples for the first time. Voids have been found using the empty sphere method which has the potential to detect and describe subtle structures in the galaxy distribution. Our approach is complementary to most other methods usually used in Large–Scale Structure studies.

The Empty Sphere Part II

Blow up a sphere in one of the interstices of a lattice until it is held rigidly. There will be no lattice points in the interior and sufficiently many on the boundary so that their convex hull is a solid figure. Such a sphere was called an empty sphere by B. N. Delone in 1924 when he introduced his method for lattice coverings [3, 4]. The circumscribed polytope is called an L-polytope. Our interest in such matters stems from the following result [6, Theorems 2.1 and 2.3]: With a list of the L-polytopes for lattices of dimension ≦n one can give a geometrical description of the possible sets of integer solutions ofwhere f satisfies the following condition (in which Z denotes the integers):

The Empty Sphere

In 1924 at the Toronto meeting of the International Congress of Mathematicians, B. N. Delone introduced his empty sphere method for lattices. We have titled our paper after this method as a tribute to his memory.We have studied the sets of integer solutions of equations of the form1where f satisfies the following condition in which Z denotes the integers,2and have resolved this problem using the theory of L-types of lattices [3, 4, 11]. We have been able to give a complete description of all such integer solutions when n ≦ 4.