Dimension-Free Lipschitz Bounds for Brenier Maps to Compact Log-Concave Targets

Abstract

Let d≥1, Λ>0, and ρ∈ℝ, let μ=𝒩(0,Λ⁻¹Id) on ℝ^d, where Id is the identity matrix, and let ν be a probability measure whose support is a convex body K of diameter D=diam(K) and whose density is proportional to e^−W, where D²W⪰−ρId. We prove that the Brenier map T from μ to ν has Lipschitz constant Lip(T)≤√(Λ/(2π))∫_−D/2^D/2e^ρt²/2 dt, which is sharp in every fixed dimension. For log-concave targets, the optimal coefficient of √Λ D is 1/√(2π). For sources with finite second moment and density proportional to e^−V for which x↦1/2⟨ Qx,x⟩−V(x) is convex, where Q is symmetric positive definite, and log-concave targets with compact support K, the Brenier potential Φ satisfies ∂_vvΦ≤ C₁√(⟨ Qv,v⟩) w_K(v) in the sense of distributions, where w_K(v):=sup_{y∈ K}⟨ y,v⟩−inf_{y∈ K}⟨ y,v⟩ is the width of K along v∈ℝ^d and C₁<0.463. This removes Kolesnikov's √d loss. For log-concave sources with D²V⪯ΛId and the target ν, the Brenier map T satisfies Lip(T)²≤Λ∫₀^D²/4e^ρt dt, so Lip(T) has the optimal exponential rate 1/8 in ρD².

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Cite

BibTeX RIS

BibTeX

@misc{gwozdz2026dimensionfree,
  title = {{Dimension-Free Lipschitz Bounds for Brenier Maps to Compact Log-Concave Targets}},
  author = {Maja Gwozdz},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2608.15906},
  eprint = {2608.15906},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.15906},
}

RIS

TY  - UNPB
TI  - Dimension-Free Lipschitz Bounds for Brenier Maps to Compact Log-Concave Targets
AU  - Maja Gwozdz
PY  - 2026
DA  - 2026/11/02
PB  - arXiv
DO  - 10.48550/arxiv.2608.15906
UR  - https://arxiv.org/abs/2608.15906
AB  - Let d≥1, Λ>0, and ρ∈ℝ, let μ=𝒩(0,Λ^−1Id) on ℝ^d, where Id is the identity matrix, and let ν be a probability measure whose support is a convex body K of diameter D=diam(K) and whose density is proportional to e^−W, where D^2W⪰−ρId. We prove that the Brenier map T from μ to ν has Lipschitz constant Lip(T)≤√(Λ/(2π))∫_−D/2^D/2e^ρt^2/2 dt, which is sharp in every fixed dimension. For log-concave targets, the optimal coefficient of √Λ D is 1/√(2π). For sources with finite second moment and density proportional to e^−V for which x↦1/2⟨ Qx,x⟩−V(x) is convex, where Q is symmetric positive definite, and log-concave targets with compact support K, the Brenier potential Φ satisfies ∂_vvΦ≤ C_1√(⟨ Qv,v⟩) w_K(v) in the sense of distributions, where w_K(v):=sup_y∈ K⟨ y,v⟩−inf_y∈ K⟨ y,v⟩ is the width of K along v∈ℝ^d and C_1<0.463. This removes Kolesnikov's √d loss. For log-concave sources with D^2V⪯ΛId and the target ν, the Brenier map T satisfies Lip(T)^2≤Λ∫_0^D^2/4e^ρt dt, so Lip(T) has the optimal exponential rate 1/8 in ρD^2.
ER  -

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