Abstract
Given semi-normsf andg on ℝn and a real number μ>0. Then the successive minima off under the constraintg≤μ are defined by λ j : = inf {λ: there existj linear independent vectors inZ n 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 λ j '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.
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Kratz, W. Sukzessive Minima mit und ohne Nebenbedingungen. Monatshefte für Mathematik 91, 275–289 (1981). https://doi.org/10.1007/BF01294768
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DOI: https://doi.org/10.1007/BF01294768