Antiprismless, or: Reducing Combinatorial Equivalence to Projective Equivalence in Realizability Problems for Polytopes

This article exhibits a 4-dimensional combinatorial polytope that has no antiprism, answering a question posed by Bernt Lindst\"om. As a consequence, any realization of this combinatorial polytope has a face that it cannot rest upon without toppling over. To this end, we provide a general method for solving a broad class of realizability problems. Specifically, we show that for any semialgebraic property that faces inherit, the given property holds for some realization of every combinatorial polytope if and only if the property holds from some projective copy of every polytope. The proof uses the following result by Below. Given any polytope with vertices having algebraic coordinates, there is a combinatorial "stamp" polytope with a specified face that is projectively equivalent to the given polytope in all realizations. Here we construct a new stamp polytope that is closely related to Richter-Gebert's proof of universality for 4-dimensional polytopes, and we generalize several tools from that proof

Transforming education policy in New Zealand: a case study analysis

This study provides a comprehensive analysis of the changing face of New Zealand education policy over the past 25 years. It highlights the phase of socio-economic trans-formation in the late 1980s and its far-reaching impact on the education system, before turning to the last two decades, in which New Zealand's education policy has been in-creasingly shaped by its system of education export, its willingness to engage in interna-tional comparison and its close cooperation with international organizations. The article also emphasizes the various domestic forces, which have shaped education policy-making. They include a unique willingness to experiment, pragmatism, and an underly-ing culture of balance and inclusion, which account for the high degree of flexibility and adaptiveness of the country's secondary and tertiary education systems. -- Diese Studie gibt einen umfassenden Überblick des Wandels in der neuseeländischen Bildungspolitik, der bereits vor ca. 25 Jahren begann. Der Fokus wird zuerst auf die Phase der sozioökonomischen Transformation Ende der 1980er Jahre und deren Auswirklungen auf das Bildungssystem gelegt. Danach werden die Reformen der letzten zwei Jahrzehnte skizziert. Sowohl im tertiären als auch im sekundären Bildungsbereich wird Bildung in diesem Zeitraum zunehmend as liberalisiertes Servicegut interpretiert, welches zudem exportierbar ist. Gleichzeitig wurde die neuseeländische Hochschulpolitik durch die Ergebnisse internationaler Leistungsvergleiche und durch enge Zusammenarbeit mit internationalen Organisationen stark geprägt. Der Artikel zeigt außerdem wie verschiedene innenpolitische Faktoren bildungspolitische Reformen in Neuseeland beschleunigt haben. Dazu gehören eine starke Bereitschaft mit neuen Lösungsansätzen zu experimentieren, ein stark ausgeprägter Pragmatismus und eine für Neuseeland charakteristische Kultur des sozialen Gleichgewichtes und Ausgleiches. Diese Faktoren erklären den hohen Grad an Flexibilität sowie die Anpassungsfähigkeit des sekundären und tertiären Bildungssystems des Landes.

Regular systems of paths and families of convex sets in convex position

In this paper we show that every sufficiently large family of convex bodies in the plane has a large subfamily in convex position provided that the number of common tangents of each pair of bodies is bounded and every subfamily of size five is in convex position. (If each pair of bodies have at most two common tangents it is enough to assume that every triple is in convex position, and likewise, if each pair of bodies have at most four common tangents it is enough to assume that every quadruple is in convex position.) This confirms a conjecture of Pach and Toth, and generalizes a theorem of Bisztriczky and Fejes Toth. Our results on families of convex bodies are consequences of more general Ramsey-type results about the crossing patterns of systems of graphs of continuous functions $f:[0,1] \to \mathbb{R}$. On our way towards proving the Pach-Toth conjecture we obtain a combinatorial characterization of such systems of graphs in which all subsystems of equal size induce equivalent crossing patterns. These highly organized structures are what we call regular systems of paths and they are natural generalizations of the notions of cups and caps from the famous theorem of Erdos and Szekeres. The characterization of regular systems is combinatorial and introduces some auxiliary structures which may be of independent interest

The Erd\H{o}s-Szekeres problem for non-crossing convex sets

We show an equivalence between a conjecture of Bisztriczky and Fejes T{\'o}th about arrangements of planar convex bodies and a conjecture of Goodman and Pollack about point sets in topological affine planes. As a corollary of this equivalence we improve the upper bound of Pach and T\'{o}th on the Erd\H{o}s-Szekeres theorem for disjoint convex bodies, as well as the recent upper bound obtained by Fox, Pach, Sudakov and Suk, on the Erd\H{o}s-Szekeres theorem for non-crossing convex bodies. Our methods also imply improvements on the positive fraction Erd\H{os}-Szekeres theorem for disjoint (and non-crossing) convex bodies, as well as a generalization of the partitioned Erd\H{o}s-Szekeres theorem of P\'{o}r and Valtr to arrangements of non-crossing convex bodies

Multilevel Venue Shopping Amid Democratic Backsliding in New European Union Member States

Recently, various Central and Eastern European countries have experienced a regression of democratic quality, often resulting in the emergence of competitive (semi‐)authoritarian regimes with an illiberal governing ideology. This has often been accompanied by a closing political space for civil society groups. Based on a survey of more than 400 Polish, Hungarian, Czech, and Slovenian interest organizations, we explore, in the context of backsliding, the conditions under which organized interests shift their lobbying activities to alternative, i.e., EU or regional levels. Our statistical analyses indicate that it is rather exclusive policy‐making in general than a lack of individual group access to domestic policy networks that motivate organizations to engage in multilevel lobbying. However, it appears that organizational self‐empowerment and inter‐group cooperation are the “name of the game.” Even under the adverse conditions of democratic backsliding, organizations that are accumulating expertise, professionalizing their operations, and cooperating with other organizations not only can sustain access to (illiberal) national governments but also branch out their operations to the European and regional levels