On Successive Minima with Constraints
Given semi-norms/, g on IR" and μ > 0 such that f+g is a norm and such that there exist n + 1 linearly independent (l.i.) vectors x 0 , . . ,,xneZ" with ί/(Χί)<μ for i = 0,. . . ,« .The successive minima of / subject to y</i are defined by /.t = inf {A>0: there exist k + 1 1. i. vectors x0,. . ,,xkeE with/(x.) < / . and g(Xj)<ß for / = 0,. . .,fc}. There always exist n + 1 l.i. vectors x 0 , . . .,x„eZ" + 1 with/(χ () = /., and 3(χ,·)<μ for ι = 0,. . Consider such a system of vectors x 0 , . . ,,x„, and let μ^ = ιMoreover, denote by V(M) the volume of a set Μ c= R " + T h e n the main result reads for /.k<X</.k+1,k = 0,. . ., η — 1 with a constant c depending on η only (assuming > 0). An important case of this inequality is the following. Given x , , . . . , . X „ E I R and let / (P) = maxIqx,· — g(P) = \q\ for P = AMS-classification: 10-E05, 10-F10