Sukzessive Minima mit und ohne Nebenbedingungen

Given semi-normsf andg on ℝn and a real number μ>0. Then the successive minima off under the constraintg≤μ are defined by λj: = inf {λ: there existj linear independent vectors inZn withf≤λ andg≤μ}. The main theorem of this paper (Lagrange multiplier theorem) states that the successive minima of a certainnorm h on ℝn (without constraints) coincide with the λj's up to bounded factors. Moreover, this norm is constructed explicitly. Using Minkowski's wellknown theorem on successive minima and our result certain inequalities on simultaneous Diophantine approximations are derived.