The Aleksandrov Monge–Ampère measure of a convex function #

Let E be a finite-dimensional real inner product space with an additive Haar measure μ, and let f : E → EReal be convex (convex real epigraph, never ⊥), with D the interior of its effective domain {x | f x ≠ ⊤}. The subgradient image of a set s is ∂f(s) = ⋃ x ∈ s, ∂f(x) (TauCeti.subgradientImage), the union of the subdifferentials of f for the inner product. The Aleksandrov Monge–Ampère measure of f is

MA_f(s) = μ(∂f(s ∩ D))

for Borel s. It gives the weak (Aleksandrov) sense in which a merely convex function solves the Monge–Ampère equation det D²f = ν. The same formula defines a measure for every μ absolutely continuous with respect to a Haar measure, such as the weighted measure s ↦ ∫_{∂f(s ∩ D)} g for μ = volume.withDensity g.

That s ↦ μ(∂f(s ∩ D)) is a measure rests on two facts.

A point y that is a subgradient at two distinct points x₁ ≠ x₂ is a point where the conjugate f⋆ has the two subgradients x₁ and x₂, so f⋆ is not differentiable there. By Rademacher's theorem for the convex function f⋆, such points form a μ-null set (TauCeti.measure_setOf_exists_ne_mem_subdifferential_eq_zero); this needs no convexity of f. Hence the subgradient images of disjoint sets are almost disjoint.
The subgradient image of a compact subset of D is compact (TauCeti.isCompact_subgradientImage). Open subsets of D are σ-compact, so their subgradient images are measurable; complements are handled by the almost-disjointness, so the subgradient image of every Borel subset of D is null measurable (TauCeti.nullMeasurableSet_subgradientImage).

A finite convex function u on an open convex set Ω is covered by extending it by ⊤ off Ω; then D = Ω and the subdifferential is the set of y with u x + ⟪x' - x, y⟫ ≤ u x' for all x' ∈ Ω (TauCeti.mongeAmpereMeasure_ite_apply).

For a convex function that is twice differentiable on D, the Aleksandrov measure is the classical one, MA_f = det (D²f) dx on D (TauCeti.mongeAmpereMeasure_eq_withDensity_det). On D the subdifferential is the gradient, so MA_f(s) = μ(∇f(s ∩ D)). This is computed by the change of variables formula along the gradient (TauCeti.withDensity_image_gradient_eq_setLIntegral): on a set where the subdifferential is the single gradient ∇f x and has a positive derivative H x, the gradient is injective where H x is nondegenerate (TauCeti.apply_sub_eq_zero_of_forall_mem_subdifferential_norm_sub_le) and maps the set where H x is degenerate to a null set. The Hessian of a convex function is positive (TauCeti.isPositive_of_hasFDerivAt_gradient).

The Aleksandrov maximum principle bounds a convex function by its Monge–Ampère mass: if u is convex on a bounded open convex Ω, continuous on the closure and nonnegative on the frontier, then max (-u x₀) 0 ^ n ≤ C diam(Ω) ^ (n - 1) dist(x₀, ∂Ω) MA_u(Ω) for x₀ ∈ Ω, with C = 2 ^ (n + 1) / μ (ball 0 1) (TauCeti.ofReal_neg_pow_mul_addHaar_ball_le_mul_mongeAmpereMeasure). Every slope p of an affine function through (x₀, u x₀) lying below u on the frontier is a subgradient of u at some point of Ω (TauCeti.exists_forall_add_inner_le_of_forall_frontier). When u ≥ 0 on the frontier and u x₀ < 0, these slopes include a convex set containing a ball of radius -u x₀ / diam Ω and a point of norm -u x₀ / dist(x₀, ∂Ω), and the volume of such a set is bounded below by Convex.ofReal_dist_mul_pow_mul_addHaar_ball_le.

Main definitions #

TauCeti.mongeAmpereMeasure μ f — the Aleksandrov Monge–Ampère measure of f; it is 0 when f is not convex or μ is not absolutely continuous with respect to a Haar measure.

Main statements #

TauCeti.measure_setOf_exists_ne_mem_subdifferential_eq_zero and TauCeti.aedisjoint_subgradientImage — almost no point is a subgradient at two distinct points, so subgradient images of disjoint sets are almost disjoint;
TauCeti.mongeAmpereMeasure_apply — MA_f(s) = μ(∂f(s ∩ D)) for measurable s;
TauCeti.mongeAmpereMeasure_add — MA_f is additive in the reference measure, among measures absolutely continuous with respect to an additive Haar measure;
TauCeti.mongeAmpereMeasure_compl_interior — MA_f is concentrated on D;
TauCeti.mongeAmpereMeasure_lt_top and TauCeti.isLocallyFiniteMeasure_comap_mongeAmpereMeasure — when μ is finite on compact sets, MA_f is finite on compact subsets of D, so it is a locally finite measure on D;
TauCeti.sigmaFinite_mongeAmpereMeasure — when μ is finite on compact sets, MA_f is σ-finite for every f;
TauCeti.mongeAmpereMeasure_ite_apply — the formula for a finite convex function on an open set;
TauCeti.mongeAmpereMeasure_ite_eq_restrict — the measure is local: restricting a convex function to an open convex subset restricts its Monge–Ampère measure;
TauCeti.withDensity_restrict_le_mongeAmpereMeasure and TauCeti.mongeAmpereMeasure_le_withDensity_restrict — comparison of a density with the Monge–Ampère measure on an open convex subset;
TauCeti.injOn_gradient_inter_setOf_det_ne_zero — the gradient is injective on the points where the subdifferential is the single gradient with an expansion of nonzero determinant;
TauCeti.withDensity_image_gradient_eq_setLIntegral — the change of variables formula ∫_{∇F(t)} g dμ = ∫_t det (H x) g (∇F x) dμ along the gradient of a convex function, on a set t where the subdifferential is the single gradient with positive derivative H x;
TauCeti.mongeAmpereMeasure_eq_withDensity_det and TauCeti.mongeAmpereMeasure_ite_eq_withDensity_det — for a twice differentiable convex function, the Aleksandrov measure is det (D²f) dx on the interior of the effective domain;
TauCeti.ofReal_neg_pow_mul_addHaar_ball_le_mul_mongeAmpereMeasure — the Aleksandrov maximum principle.

References #

A. D. Aleksandrov, Dirichlet's problem for the equation Det ‖z_{ij}‖ = φ(z₁, …, zₙ, z, x₁, …, xₙ). I, Vestnik Leningrad. Univ. Ser. Mat. Meh. Astr. 13 (1958), 5–24.
C. E. Gutiérrez, The Monge–Ampère Equation, 2nd ed., Progress in Nonlinear Differential Equations and Their Applications 89, Birkhäuser, 2016, §1.1, in particular Theorem 1.1.13, and §1.4 for the Aleksandrov maximum principle.
A. Figalli, The Monge–Ampère Equation and Its Applications, Zurich Lectures in Advanced Mathematics, EMS, 2017, §2.1, in particular Theorem 2.3.

The Aleksandrov Monge–Ampère measure of a convex function #

Main definitions #

Main statements #

References #

Change of variables along the gradient #

The Aleksandrov maximum principle #