Comparing measures on symmetric convex sets, and products #
Vaaler's cube-slicing theorem rests on a comparison between two measures on symmetric convex
sets. If γ is the Gaussian measure exp (-π ‖x‖ ^ 2) dx on ℝⁿ = ℝ^{n₁} × ⋯ × ℝ^{n_k} and B
is a product of balls B₁ × ⋯ × B_k of volume one, then γ A ≤ volume (A ∩ B) for every
symmetric convex body A (Bombieri–Gubler, Lemma C.3.7). The case of a single ball is a
comparison along rays. The general case follows from it by induction on the number of factors,
and this file proves that induction step for arbitrary measures.
Say that μ is dominated by ν on symmetric convex sets if μ s ≤ ν s for every measurable
convex s with -s = s. By the layer-cake formula, μ is then dominated by ν on even
log-concave functions as well, since their superlevel sets are symmetric and convex. The
domination passes to products. For a symmetric convex s ⊆ E × F, integrate first over F: the
slice measures x ↦ μ₂ (Prod.mk x ⁻¹' s) form an even log-concave function by the
Prékopa–Leindler inequality for μ₂, so their μ₁-integral is at most their ν₁-integral.
After exchanging the order of integration, the slices y ↦ ν₁ ((·, y) ⁻¹' s) are even and
log-concave by the Prékopa–Leindler inequality for ν₁, and their μ₂-integral is at most their
ν₂-integral. Iterating gives the statement for finite products.
For the Gaussian-against-ball comparison, each factor measure is a log-concave density times
Lebesgue measure. These have the Prékopa–Leindler property by
MeasureTheory.Measure.HasPrekopaLeindler.withDensity and
MeasureTheory.Measure.HasPrekopaLeindler.restrict.
Main results #
TauCeti.lintegral_le_lintegral_of_forall_convex: domination on symmetric convex sets implies domination on even log-concave functions.MeasureTheory.MeasurePreserving.measure_le_of_forall_convex: domination transfers along a linear map preserving both measures.TauCeti.prod_apply_le_prod_apply_of_forall_convex: domination on symmetric convex sets passes to product measures.TauCeti.pi_apply_le_pi_apply_of_forall_convex: the same for finite products.
References #
- E. Bombieri and W. Gubler, Heights in Diophantine Geometry, Cambridge University Press, 2006, Appendix C.3: Lemma C.3.4 (marginals of log-concave functions) and the proof of Lemma C.3.7.
- J. D. Vaaler, A geometric inequality with applications to linear forms, Pacific J. Math. 83 (1979), 543–553.
Domination on symmetric convex sets gives domination on even log-concave functions. If
μ s ≤ ν s for every measurable convex set s with -s = s, then
∫⁻ x, f x ∂μ ≤ ∫⁻ x, f x ∂ν for every measurable, even, log-concave f : E → ℝ≥0∞. Each
superlevel set of f is symmetric and convex, and the layer-cake formula integrates their
measures.
Domination on symmetric convex sets transfers along linear maps. If a linear map L
sends μ to μ' and ν to ν', and μ is dominated by ν on symmetric convex sets, then so
is μ' by ν': the preimage under L of a symmetric convex set is symmetric and convex.
Domination on symmetric convex sets passes to products (the induction step of
Bombieri–Gubler, Lemma C.3.7). Let μ₁ be dominated by ν₁ on symmetric convex subsets of E,
and μ₂ by ν₂ on symmetric convex subsets of F. Suppose that ν₁ and μ₂ are
negation-invariant and have the Prékopa–Leindler property at every weight. Then
μ₁.prod μ₂ s ≤ ν₁.prod ν₂ s for every measurable convex set s ⊆ E × F with -s = s.
Applied to the Gaussian measure against the uniform measure on a ball of volume one in each factor, this reduces the comparison for a product of balls to the comparison for one ball.
Domination on symmetric convex sets passes to finite products (Bombieri–Gubler, the
induction in the proof of Lemma C.3.7). Let μ i be dominated by ν i on symmetric convex
subsets of α i for each i, where all these measures are σ-finite, negation-invariant and have
the Prékopa–Leindler property at every weight. Then Measure.pi μ s ≤ Measure.pi ν s for every
measurable convex set s ⊆ ∀ i, α i with -s = s.