arXiv subjects

Krzysztof J. Ciosmak

Publications and source records attributed to Krzysztof J. Ciosmak.

14 recordsLinked to original sources

arXiv2026

Absolutely minimising Lipschitz extensions between Hilbert spaces

Every Lipschitz map from a closed subset of a real Hilbert space into a real Hilbert space has an absolutely minimising Lipschitz extension: an extension whose Lipschitz constant on every open subset of the complement, bounded or not, equals the Lipschitz constant of its restriction to the boundary of that subset. Neither space need be separable. This answers a question of Naor and Sheffield. For targets of dimension at least two we prove a density theorem on connected domains: every optimal extension of a boundary datum can be approximated, uniformly on compact sets, by extensions whose Lipschitz constant on every nonempty open subset, and on its boundary, equals the constant of the datum. Consequently, in contrast with the scalar case, such a datum has a unique absolutely minimising extension exactly when its optimal extension is unique, and absolutely minimising extensions need not be closed under uniform convergence, even for fixed smooth data on a planar disk. The extensions are built by finite piecewise isometric interpolation: on nested polyhedra with dense union they are piecewise affine and preserve distances in at least two directions on each simplex, a structure described through leaves of 1-Lipschitz maps. The boundary constraints are handled by a midpoint argument for admissible values, and a Kuratowski-Zorn argument removes the separability assumption while preserving previously prescribed values. We also characterise the bounded linear maps that are absolutely minimising extensions of all data determining their operator norm.

math.FA↗

arXiv2026

Measure contraction property on isometric leaves and monotone fibres

For finite measures with positive densities on convex Euclidean supports, we prove that $MCP(κ,N)$ passes with unchanged parameters to almost every isometric leaf of an arbitrary nonexpansive map. The proof rests on a sharp contraction inequality for geometric conditional densities, with exponent equal to the leaf codimension. The inherited dimension parameter is optimal. A total-variation limit on resolvent graphs extends the result to inverse fibres of maximal monotone relations, including convex gradient fibres. We also disprove Klartag's curvature-dimension inheritance conjecture by a firmly nonexpansive example in dimension three and a gradient example in dimension four. In codimension one, affinity of the geometric density yields curvature-dimension inheritance. The first example also gives failure on monotone fibres. Both constructions admit arbitrarily large curvature loss, including for a fixed Gaussian ambient measure on families of leaves of positive quotient measure.

math.MG↗

arXiv2026

Kirszbraun extensions preserving uniform distance in Hilbert spaces

Let $X$ be a subset of a real Hilbert space and let $v\colon X\to Y$, where $Y$ is a real Hilbert space. We prove that the following conditions are equivalent: whenever $A\subset X$, $ρ\geq0$, and $u\colon A\to Y$ is $1$-Lipschitz with $\left\lVert u(x)-v(x)\right\lVert\leqρ$ for $x\in A$, there is a $1$-Lipschitz extension $\widetilde u\colon X\to Y$ with $\left\lVert \widetilde u(x)-v(x)\right \lVert\leqρ$ for $x\in X$; and for every $1\leq k\leq\dim Y$, $$ \left\lVert v(x_0)-\sum_{i=1}^k t_i v(x_i)\right\lVert \leq \left\lVert x_0-\sum_{i=1}^k t_i x_i \right \lVert$$ whenever $x_0,\ldots,x_k\in X$, $t_1,\ldots,t_k\geq0$, and $\sum_{i=1}^k t_i=1$. Previous necessity results required $\dim Y\leq3$ or convexity of $X$. For finite-dimensional targets, an application gives an exact data processing characterisation for a finite branching hierarchy connecting Wasserstein and barycentric weak transport. If $Y$ is infinite-dimensional or $\dim\operatorname{Aff}X+1\leq\dim Y$, we also obtain a lifting theorem for convex Lipschitz functions and transfer convex Poincaré inequalities without increasing the constant.

math.FA↗

arXiv2026

Localisation for the second-order Beckmann problem: bimartingale couplings and leaf decompositions

We develop a second-order localisation theory for optimal transport based on leaf decompositions and bimartingale couplings. It provides a second-order analogue of the classical decomposition of optimal transport into transport rays and monotone couplings: transport rays are replaced by the leaves of the $1$-Lipschitz derivative map $Du$ of an optimal dual potential $u\in C^{1,1}(\mathbb R^n)$, while monotone couplings are replaced by bimartingale couplings. We apply this framework to the three-marginal optimal transport problem introduced by Bolbotowski and Bouchitté, whose relaxation is the second-order Beckmann problem. We introduce bimartingale couplings and characterise their existence through a convex-concave order condition. This yields a generalisation of Strassen's theorem from convex order to the convex-concave setting. Equivalently, the dual problem associated with the second-order Beckmann problem admits an optimiser whose derivative is an isometry. For absolutely continuous measures with common barycentre, assuming the existence of an optimal plan with absolutely continuous third marginal, we prove that every optimal plan decomposes into a family of problems on the leaves of $Du$. On each leaf, all optimal plans are completely described by bimartingale couplings between the corresponding conditional measures. Without this absolute continuity assumption, we show that the leaf decomposition persists in a more general form: optimal plans are mixtures of plans concentrated on triples $(x,y,z)$ satisfying $x\in \mathcal{S}_1$, $y\in \mathcal{S}_2$, and $z\in \mathcal{S}_1\cap \mathcal{S}_2$, where $\mathcal{S}_1$ and $\mathcal{S}_2$ are neighbouring leaves of $Du$.

math.OC↗

arXiv2025

Cartan-Thullen theorem and Levi problem in context of generalised convexity

We demonstrate that the Cartan-Thullen theorem and its generalisation to the context of generalised convexity, which we establish herein, can be regarded as consequences of the classical theorems of functional analysis: the Banach-Steinhaus theorem and the Banach-Alaoglu theorem. Furthermore, we characterise the domains of holomorphy, and their generalisations, as the spaces that are complete, or as the spaces exhaustible by suitably defined polytopes. We also provide an abstract analogue of the Levi problem and its elementary resolution. Our results allow for a novel characterisation of Stein spaces as the holomorphically complete spaces, as well as a proof that the Bremermann-Lelong lemma is equivalent to the positive answer to the Levi problem. Another contribution of ours is the introduction of the analogues of the notions of the complex analysis to the setting of generalised convexity.

math.CV↗

arXiv2024

Continuity of extensions of Lipschitz maps and of monotone maps

Let $X$ be a subset of a Hilbert space. We prove that if $v\colon X\to \mathbb{R}^m$ is such that \begin{equation*} \Big\lVert v(x)-\sum_{i=1}^m t_iv(x_i)\Big\rVert\leq \Big\lVert x-\sum_{i=1}^m t_ix_i\Big\rVert \end{equation*} for all $x,x_1,\dotsc,x_m\in\mathbb{R}^m$ and all non-negative $t_1,\dotsc,t_m$ that add up to one, then for any $1$-Lipschitz $u\colon A\to\mathbb{R}^m$, with $A\subset X$, there exists a $1$-Lipschitz extension $\tilde{u}\colon X\to\mathbb{R}^m$ of $u$ such that the uniform distance on $X$ between $\tilde{u}$ and $v$ is the same as the uniform distance on $A$ between $u$ and $v$. Moreover, if either $m\in \{1,2,3\}$ or $X$ is convex, we prove the converse: we show that a map $v\colon X\to\mathbb{R}^m$ that allows for a $1$-Lipschitz, uniform distance preserving extension of any $1$-Lipschitz map on a subset of $X$ also satisfies the above set of inequalities. We also prove a similar continuity result concerning extensions of monotone maps. Our results hold true also for maps taking values in infinite-dimensional spaces.

math.FA↗

arXiv2024

Localisation for constrained transports I: theory

We investigate an analogue of the irreducible convex paving in the context of generalised convexity. Consider two Radon probability measures $μ,ν$ ordered with respect to a cone $\mathcal{F}$ of functions on $Ω$ stable under maxima. Under the assumption that any $\mathcal{F}$-transport between $μ$ and $ν$ is local, we establish the existence of the finest partitioning of $Ω$, depending only on $μ,ν$ and the cone $\mathcal{F}$, into $\mathcal{F}$-convex sets, called irreducible components, such that any $\mathcal{F}$-transport between $μ$ and $ν$ must adhere to this partitioning. Furthermore, we demonstrate that a set, whose sections are contained in the corresponding irreducible components, is a polar set with respect to all $\mathcal{F}$-transports between $μ$ and $ν$ if and only if it is a polar set with respect to all transports. This provides an affirmative answer to a generalisation of a conjecture proposed by Obłój and Siorpaes regarding polar sets in the martingale transport setting. Among our contributions is also a generalisation of the Strassen's theorem to the setting of generalised convexity

math.PR↗

arXiv2024

Localisation for constrained transports II: applications

We present a range of applications of localisation for constrained transports for pairs of probability measures in order with respect to a lattice cone. These examples comprise irreducible convex paving for martingale transports in infinite-dimensional spaces, irreducible convex paving for submartingale transports, localisation of the Monge--Kantorovich problem, pavings for harmonic transports, pavings for solutions of martingale problems. Furthermore, we consider approximating martingale transports and characterise the polar sets in this setting, identifying them as the trivial polar sets -- the sets whose projections have vanishing measures. This provides a negative answer to a question of Cox

math.PR↗

arXiv2023

Applications of Strassen's theorem and Choquet theory to optimal transport problems, to uniformly convex functions and to uniformly smooth functions

We provide a unifying interpretation of various optimal transport problems as a minimisation of a linear functional over the set of all Choquet representations of a given pair of probability measures ordered with respect to a certain convex cone of functions. This allows us to provide novel proofs of duality formulae. Among our tools is Strassen's theorem. We provide new formulations of the primal and the dual problem for martingale optimal transport employing a novel representation of the set of extreme points of probability measures in convex order on Euclidean space. We exhibit a link to uniformly convex and uniformly smooth functions and provide a new characterisation of such functions. We introduce a notion of martingale triangle inequality. We show that Kantorovich--Rubinstein duality bears an analogy in the martingale setting employing the cost functions that satisfy the inequality.

math.FA↗

arXiv2021

Matrix Hölder's inequality and divergence formulation of optimal transport of vector measures

We characterise equality cases in matrix Hölder's inequality and develop a divergence formulation of optimal transport of vector measures. As an application, we reprove the representation formula for measures in the polar cone to monotone maps. We generalise the last result to a wide class of polar cones, including polar cones to tangent cones to the unit ball in the space of differentiable functions and in the Sobolev spaces.

math.FA↗

arXiv2021

Continuity of extensions of Lipschitz maps

We establish the sharp rate of continuity of extensions of $\mathbb{R}^m$-valued $1$-Lipschitz maps from a subset $A$ of $\mathbb{R}^n$ to a $1$-Lipschitz maps on $\mathbb{R}^n$. We consider several cases when there exists a $1$-Lipschitz extension with preserved uniform distance to a given $1$-Lipschitz map. We prove that if $m>1$ then a given map is $1$-Lipschitz and affine if and only if such distance preserving extension exists for any $1$-Lipschitz map defined on any subset of $\mathbb{R}^n$. This shows a striking difference from the case $m=1$, where any $1$-Lipschitz function has such property. Another example where we prove it is possible to find an extension with the same Lipschitz constant and the same uniform distance to another Lipschitz map $v$ is when the difference between the two maps takes values in a fixed one-dimensional subspace of $\mathbb{R}^m$ and the set $A$ is geodesically convex with respect to a Riemannian pseudo-metric associated with $v$.

math.FA↗

arXiv2021

Leaves decompositions in Euclidean spaces

We partly extend the localisation technique from convex geometry to the multiple constraints setting. For a given $1$-Lipschitz map $u\colon\mathbb{R}^n\to\mathbb{R}^m$, $m\leq n$, we define and prove the existence of a partition of $\mathbb{R}^n$, up to a set of Lebesgue measure zero, into maximal closed convex sets such that restriction of $u$ is an isometry on these sets. We consider a disintegration, with respect to this partition, of a log-concave measure. We prove that for almost every set of the partition of dimension $m$, the associated conditional measure is log-concave. This result is proven also in the context of the curvature-dimension condition for weighted Riemannian manifolds. This partially confirms a conjecture of Klartag.

math.MG↗

arXiv2021

Optimal transport of vector measures

We develop and study a theory of optimal transport for vector measures. We resolve in the negative a conjecture of Klartag, that given a vector measure on Euclidean space with total mass zero, the mass of any transport set is again zero. We provide a counterexample to the conjecture. We generalise the Kantorovich--Rubinstein duality to the vector measures setting. Employing the generalisation, we answer the conjecture in the affirmative provided there exists an optimal transport with absolutely continuous marginals of its total variation.

math.FA↗

arXiv2020

Leaves decompositions in Euclidean spaces and optimal transport of vector measures

For a given $1$-Lipschitz map $u\colon\mathbb{R}^n\to\mathbb{R}^m$ we define a partition, up to a set of Lebesgue measure zero, of $\mathbb{R}^n$ into maximal closed convex sets such that restriction of $u$ is an isometry on these sets. We consider a disintegration, with respect to this partition, of a log-concave measure. We prove that for almost every set of the partition of dimension $m$, the associated conditional measure is log-concave. This result is proven also in the context of the curvature-dimension condition $CD(κ,N)$ for weighted Riemannian manifolds. This partially confirms a conjecture of Klartag. We provide a counterexample to another conjecture of Klartag that, given a vector measure on $\mathbb{R}^n$ with total mass zero, the conditional measures, with respect to partition obtained from a certain $1$-Lipschitz map, also have total mass zero. We develop a theory of optimal transport for vector measures and use it to answer the conjecture in the affirmative provided a certain condition is satisfied.

math.MG↗