Affine renormalization of the Monge–Ampère measure #
Adding the affine function x ↦ ⟪x, q⟫ + c to a function f : E → EReal translates every
subdifferential, and hence every subgradient image, by q
(TauCeti.subgradientImage_add_affine). It also preserves convexity of the real epigraph and
the effective domain, so the Aleksandrov Monge–Ampère measure with respect to an additive Haar
measure is unchanged.
This invariance permits the standard normalization of a convex function by subtracting a supporting plane before studying its sections.
Precomposing with an affine equivalence A of linear part L changes coordinates in the domain:
the subgradients of f ∘ A at x are the images under the adjoint L* of the subgradients of
f at A x (TauCeti.subdifferential_comp_affineEquiv). Since L* multiplies Haar measure by
|det L|, the Monge–Ampère measure transforms by MA_{f ∘ A}(s) = |det L| MA_f(A '' s). So a
lower bound MA_f ≥ λ μ on a set s becomes MA_{f ∘ A} ≥ λ |det L|² μ on A⁻¹(s), since
μ(A '' t) = |det L| μ(t): this is how an estimate proved for a normalized convex domain
transfers to an arbitrary one.
Main statements #
TauCeti.mongeAmpereMeasure_mul_of_posgives the homogeneity of the Monge–Ampère measure under positive scaling of an extended-real potential,TauCeti.mongeAmpereMeasure_ite_mul_of_posis its form for a real function on a set, andMeasureTheory.Measure.mongeAmpereMeasure_ite_mul_add_of_posis its form for scaling followed by a constant shift.TauCeti.mongeAmpereMeasure_add_affineproves invariance of the Aleksandrov Monge–Ampère measure under addition of an affine function, andTauCeti.mongeAmpereMeasure_ite_add_linearMapis its form for a finite convex function on a set and an affine function given by a linear functional.TauCeti.mongeAmpereMeasure_comp_affineEquiv_applyandTauCeti.mongeAmpereMeasure_comp_affineEquivgive the change of variables formula for the Monge–Ampère measure under an affine equivalence of the domain, andTauCeti.mongeAmpereMeasure_ite_comp_symm_applyis its form for a finite function on a set transported to the image of the set.
References #
- C. E. Gutiérrez, The Monge–Ampère Equation, 2nd ed., Progress in Nonlinear Differential Equations and Their Applications 89, Birkhäuser, 2016, §1.1.
Precomposing with an affine equivalence A maps the subdifferential at A x onto the
subdifferential at x by the adjoint of the linear part of A.
Affine invariance of the Aleksandrov Monge–Ampère measure. Adding an affine function
x ↦ ⟪x, q⟫ + c to f : E → EReal does not change its Monge–Ampère measure with respect to an
additive Haar measure.
Affine invariance for a finite convex function on a set. Adding a linear functional ℓ
and a constant to a real function u, extended by ⊤ off U, does not change its Monge–Ampère
measure with respect to an additive Haar measure. This is the form in which a supporting plane
y ↦ u x + ℓ (y - x) is subtracted from u to normalize a section.
Multiplying an extended-real potential by a positive scalar c multiplies its Aleksandrov
Monge–Ampère measure with respect to additive Haar measure by c ^ finrank ℝ E.
Positive scaling of a real potential on a set, extended by ⊤ off that set.
Positive scaling and a constant shift of a real potential on a set, extended by ⊤ off that
set: the Monge–Ampère measure is multiplied by c ^ finrank ℝ E.
Change of variables for the Aleksandrov Monge–Ampère measure. For an affine equivalence
A of E with linear part L and a measurable set s,
MA_{f ∘ A}(s) = |det L| MA_f(A '' s) for the Monge–Ampère measures with respect to an additive
Haar measure.
Transport of the Monge–Ampère measure of a function on G to the function v ∘ T⁻¹ on
T '' G = T⁻¹ ⁻¹' G: the measure of s is |det L| times the Monge–Ampère measure of v on
G of T⁻¹ '' s, where L is the linear part of T⁻¹.
Change of variables for the Aleksandrov Monge–Ampère measure, as an identity of measures:
for an affine equivalence A of E with linear part L, the Monge–Ampère measure of f ∘ A
is |det L| times the pullback of that of f along A.