The Sine Transform of Isotropic Measures

Sharp isoperimetric inequalities for the sine transform of even isotropic measures are established. The corresponding reverse inequalities are obtained in an asymptotically optimal form. These new inequalities have direct applications to strong volume estimates for convex bodies from data about their sections or projections

Optimal and better transport plans

We consider the Monge-Kantorovich transport problem in a purely measure theoretic setting, i.e. without imposing continuity assumptions on the cost function. It is known that transport plans which are concentrated on c-monotone sets are optimal, provided the cost function c is either lower semi-continuous and finite, or continuous and may possibly attain the value infty. We show that this is true in a more general setting, in particular for merely Borel measurable cost functions provided that {c=infty} is the union of a closed set and a negligible set. In a previous paper Schachermayer and Teichmann considered strongly c-monotone transport plans and proved that every strongly c-monotone transport plan is optimal. We establish that transport plans are strongly c-monotone if and only if they satisfy a "better" notion of optimality called robust optimality.Comment: 25 page

Hartman measurable sets and functions

Wir definieren Hartman Meßbarkeit von Funktionen wie folgt: Eine auf der (nichtkompakten) topologischen Gruppe $G$ definierte Funktion f: G o \C ist Hartman meßbar, wenn es eine Riemann integrierbare Funktion F: C o \C gibt (eine Funktion ist Riemann integrierbar, wenn die Menge ihrer Unstetigkeitsstellen eine Nullmenge bezüglich des Haarschen Maßes ist) und eine Gruppenkompaktifizierung $(\iota,C)$, sodass $f=F\circ \iota$. Insbesondere interessieren wir uns für die Struktur der Menge aller Gruppenkompaktifizierungen $(\iota,C)$ von $G$, sodass eine gegebene Hartman meßbare Funktion $f$ als $F\circ\iota$ mit einer auf $C$ definierten Riemann integrierbaren Funktion $F$ dargestellt (wir sagen auch realisiert) werden kann. Ist $G$ eine LCA Gruppe mit separablem Dual, so ist eine solche Realisierung stets schon auf einer metrisierbaren Gruppe $C$ möglich. In wichtigen Spezialfällen läßt sich eine Realisierung auch mit Hilfe des Fourierspektrums von $f$ angeben. Jede fastperiodische Funktion ist insbesondere auch Hartman meßbar. Eine bekannte und gut untersuchte Verallgemeinerung der fastperiodischen Funktionen stellen die schwach fastperiodischen Funktionen dar. Wir zeigen, dass es i.a. Hartman meßbare Funktionen gibt, welche nicht schwach fastperiodisch sind. $C_0$-Funktionen dagegen, i.e. Funktionen die im Unendlichen verschwinden, sind stets Hartman meßbar. Wir beschäftigen uns außerdem mit dem Zusammenhang zwischen Fouriertransformation von Maßen und Hartman Meßbarkeit.A subset $H$ of $\Z$, or more generally, of a discrete group $G$ is called Hartman measurable, if $H=\iota {-1}(M)$ for some continuous homomorphism $\iota: G o C$, $C=\overline{\iota(G)}$ a compact group, and $M\subset C$ with topological boundary $\partial M$ of Haar measure $0$. This concept turned out to be useful in the context of coding $n\alpha$-sequences and related group-rotations. There exists a unique translation invariant finitely additive measure on the Boolean algebra of Hartman measurable subsets. We extend this concept from sets to functions via a process similar to the passage from measurable sets to measurable functions. The relation between this class of Hartman-measurable functions and the class of almost periodic functions is comparable to the relation between Riemann integrable functions to continuous functions. We present results about the extent of Hartman measurable functions and establish a connection between Hartman measurability and concepts such as (weak) almost periodicity. We use methods from strucure theory of LCA groups and fourier analysis.6

Global impact of the first coronavirus disease 2019 (COVID-19) pandemic wave on vascular services

This online structured survey has demonstrated the global impact of the COVID-19 pandemic on vascular services. The majority of centres have documented marked reductions in operating and services provided to vascular patients. In the months during recovery from the resource restrictions imposed during the pandemic peaks, there will be a significant vascular disease burden awaiting surgeons. One of the most affected specialtie

Documenting the Recovery of Vascular Services in European Centres Following the Initial COVID-19 Pandemic Peak: Results from a Multicentre Collaborative Study

Objective: To document the recovery of vascular services in Europe following the first COVID-19 pandemic peak. Methods: An online structured vascular service survey with repeated data entry between 23 March and 9 August 2020 was carried out. Unit level data were collected using repeated questionnaires addressing modifications to vascular services during the first peak (March - May 2020, "period 1"), and then again between May and June ("period 2") and June and July 2020 ("period 3"). The duration of each period was similar. From 2 June, as reductions in cases began to be reported, centres were first asked if they were in a region still affected by rising cases, or if they had passed the peak of the first wave. These centres were asked additional questions about adaptations made to their standard pathways to permit elective surgery to resume. Results: The impact of the pandemic continued to be felt well after countries' first peak was thought to have passed in 2020. Aneurysm screening had not returned to normal in 21.7% of centres. Carotid surgery was still offered on a case by case basis in 33.8% of centres, and only 52.9% of centres had returned to their normal aneurysm threshold for surgery. Half of centres (49.4%) believed their management of lower limb ischaemia continued to be negatively affected by the pandemic. Reduced operating theatre capacity continued in 45.5% of centres. Twenty per cent of responding centres documented a backlog of at least 20 aortic repairs. At least one negative swab and 14 days of isolation were the most common strategies used for permitting safe elective surgery to recommence. Conclusion: Centres reported a broad return of services approaching pre-pandemic "normal" by July 2020. Many introduced protocols to manage peri-operative COVID-19 risk. Backlogs in cases were reported for all major vascular surgeries