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Successive Minima


The successive minima lambda_1(K,Lambda)<=...<=lambda_n(K,Lambda) of an origin-symmetric convex body K subset R^n with respect to a full-rank point lattice Lambda measure the scales needed to obtain independent lattice points. The ith successive minimum is the least lambda>0 such that lambdaK contains i linearly independent points of Lambda. The convex body is assumed compact, with the origin in its interior (Betke et al. 1993, Aliev and Henk 2023).

For a nonsymmetric convex body K, lattice-point inequalities often use the successive minima of its symmetrized difference body (K-K)/2. Here K-K={x-y:x,y in K} denotes a Minkowski sum with -K, rather than set-theoretic subtraction.


See also

Convex Body, Lattice Point, Point Lattice

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References

Aliev, I. and Henk, M. "Minkowski's Successive Minima in Convex and Discrete Geometry." Commun. Math. 31, 35-59, 2023. https://doi.org/10.46298/cm.11155.Betke, U.; Henk, M.; and Wills, J. M. "Successive-Minima-Type Inequalities." Discrete Comput. Geom. 9, 165-175, 1993. https://doi.org/10.1007/BF02189316.

Cite this as:

Weisstein, Eric W. "Successive Minima." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SuccessiveMinima.html

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