A finite absolute-maximizing family almost norms the unit sphere

MathlibAnnex.Satellite.exists_goodSatellitePackage

theorem

Chooses the inverse bound, net and weight before fixing one satellite family.

Statement

Let and use reference coordinates with the sup norm. Let be a continuous seminorm with specified constants such that Thus is a norm. We use for the model distance and for reference-coordinate estimates; the two norms are allowed to coincide. A -dual contraction is a continuous real linear functional satisfying for every . Let be the standard coordinate vectors. For an ordered family of continuous real linear functionals on , put For a real number , define the attained maximum and near-maximal set by Admissibility is measured with ; the matrices are written in the fixed reference coordinates. Suppose and . The conclusion chooses a number with the uniform bound then a finite center set and a weight . More precisely, put Index the coefficients by and put . For a functional , let replace row by . A configuration consists of a base frame and a family of additional continuous real linear functionals , called satellites. Its polynomial, admissible set and coefficient budget are For the selected absolute maximizer , that is, the same choices satisfy and

Assumptions

The real norm model with its comparison constants, , and are the only assumptions. Neither , , a nonempty sphere nor nonempty centers are assumed. The number , its uniform bound, the finite net, and the weight are outputs, not further assumptions.

Conclusion

One finite family, obtained from the selected absolute-maximizing configuration after fixing the net and weight, works for every unit vector. The center/index may depend on the unit vector. Positivity of the weight and its strict budget inequality are also outputs.

Proof route

Fix a uniform inverse bound, choose a finite coefficient net, choose a dominating weight, maximize the absolute polynomial, and then detect each unit vector using that same family.

A common inverse bound consists of a number and the uniform estimate It is fixed independently of the particular frame and coefficient. When , every such frame has nonzero determinant. The annulus itself is the coefficient annulus. The strict epsilon version, finite-net absolute-maximizer conclusion and weight choice for a given net preserve this order of choices. As a contextual connection, define with the sup norm on its finite coordinates. the upper bound for that coordinate map uses admissibility to give , while one satellite coordinate bounds the whole map from below gives on the unit sphere. For , apply the latter to and use linearity to get , as in the lower bound for every nonzero vector. This only explains the assigned coordinate-map context; no later classification is asserted.

theorem exists_goodSatellitePackage
    {n : ℕ} (M : NormModel n) {η ε : ℝ}
    (hη : 0 < η) (hηD : η < detMax M) (hε : 0 < ε) :
    ∃ H : NearMaxInverseBound M η,
      ∃ C : FiniteCoefficientNet (n := n) H.boundConstant
        (satelliteRadiusNNReal ε H.boundConstant hε H.boundConstant_pos),
        ∃ weight : ℝ,
          0 < weight ∧
          satelliteBudget M (Internal.centerValue C) < weight * η ∧
          ∀ x : Coord n, M.p x = 1 →
            ∃ a : C.centers,
              1 - ε <
                |(maxSatelliteConfiguration M C.centers
                    weight (Internal.centerValue C)).2 a x|

private abbrev Coord (n : ℕ) := Fin n → ℝ
private abbrev NormModel (n : ℕ) := EquivalentSeminorm (Coord n)

private abbrev NearMaxInverseBound {n : ℕ} (M : NormModel n) := DeterminantFrame.NearMaxInverseBound (modelBasis M)

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