0000000000154347
AUTHOR
Timo Schultz
showing 6 related works from this author
Optimal transport maps on Alexandrov spaces revisited
2018
We give an alternative proof for the fact that in $n$-dimensional Alexandrov spaces with curvature bounded below there exists a unique optimal transport plan from any purely $(n-1)$-unrectifiable starting measure, and that this plan is induced by an optimal map.
Existence of optimal transport maps in very strict CD(K,∞) -spaces
2018
We introduce a more restrictive version of the strict CD(K,∞) -condition, the so-called very strict CD(K,∞) -condition, and show the existence of optimal maps in very strict CD(K,∞) -spaces despite the possible lack of uniqueness of optimal plans. peerReviewed
Pintojen perusryhmistä
2015
Tässä tutkielmassa osoitetaan ennestään tunnettu pintoihin liittyvä tulos, jonka mukaan epäkompaktin pinnan perusryhmä on vapaa. Todistus pohjautuu tietoon siitä, että jokaisella pinnalla on olemassa niin sanottu kolmiointi. Pinnan kolmiointia hyödyntäen pinta tyhjennetään sopivilla sisäkkäisillä kompakteilla reunallisilla pinnoilla siten, että pinnan perus ryhmä saadaan näiden kompaktien reunallisten pintojen sisäkkäisten pe rusryhmien yhdisteenä. Kompakti reunallinen pinta osoitetaan homotopia ekvivalentiksi graafin kanssa deformaatioretraktoimalla reunallinen pinta graafiksi reunallisen pinnan kolmiointia hyödyntäen. Koska homotopiaekvi valenttien avaruuksien perusryhmät ovat isomorfiset…
A Density Result for Homogeneous Sobolev Spaces on Planar Domains
2018
We show that in a bounded simply connected planar domain $\Omega$ the smooth Sobolev functions $W^{k,\infty}(\Omega)\cap C^\infty(\Omega)$ are dense in the homogeneous Sobolev spaces $L^{k,p}(\Omega)$.
On one-dimensionality of metric measure spaces
2019
In this paper, we prove that a metric measure space which has at least one open set isometric to an interval, and for which the (possibly non-unique) optimal transport map exists from any absolutely continuous measure to an arbitrary measure, is a one-dimensional manifold (possibly with boundary). As an immediate corollary we obtain that if a metric measure space is a very strict $CD(K,N)$ -space or an essentially non-branching $MCP(K,N)$-space with some open set isometric to an interval, then it is a one-dimensional manifold. We also obtain the same conclusion for a metric measure space which has a point in which the Gromov-Hausdorff tangent is unique and isometric to the real line, and fo…
Equivalent definitions of very strict $CD(K,N)$ -spaces
2023
We show the equivalence of the definitions of very strict $CD(K,N)$ -condition defined, on one hand, using (only) the entropy functionals, and on the other, the full displacement convexity class $\mathcal{DC}_N$. In particular, we show that assuming the convexity inequalities for the critical exponent implies it for all the greater exponents. We also establish the existence of optimal transport maps in very strict $CD(K,N)$ -spaces with finite $N$.