Asymptotic bounds on the combinatorial diameter of random polytopes

The combinatorial diameter $\operatorname{diam}(P)$ of a polytope $P$ is the maximum shortest path distance between any pair of vertices. In this paper, we provide upper and lower bounds on the combinatorial diameter of a random "spherical" polytope, which is tight to within one factor of dimension when the number of inequalities is large compared to the dimension. More precisely, for an $n$-dimensional polytope $P$ defined by the intersection of $m$ i.i.d.\ half-spaces whose normals are chosen uniformly from the sphere, we show that $\operatorname{diam}(P)$ is $\Omega(n m^{\frac{1}{n-1}})$ and $O(n^2 m^{\frac{1}{n-1}} + n^5 4^n)$ with high probability when $m \geq 2^{\Omega(n)}$. For the upper bound, we first prove that the number of vertices in any fixed two dimensional projection sharply concentrates around its expectation when $m$ is large, where we rely on the $\Theta(n^2 m^{\frac{1}{n-1}})$ bound on the expectation due to Borgwardt [Math. Oper. Res., 1999]. To obtain the diameter upper bound, we stitch these ``shadows paths'' together over a suitable net using worst-case diameter bounds to connect vertices to the nearest shadow. For the lower bound, we first reduce to lower bounding the diameter of the dual polytope $P^\circ$, corresponding to a random convex hull, by showing the relation $\operatorname{diam}(P) \geq (n-1)(\operatorname{diam}(P^\circ)-2)$. We then prove that the shortest path between any ``nearly'' antipodal pair vertices of $P^\circ$ has length $\Omega(m^{\frac{1}{n-1}})$

Towards EU-MENA Shared Prosperity

With a strong conviction that fruitful policy dialogue should be primarily anchored in sound policy research, Bruegel and PCNS researchers have tackled issues of utmost importance to all shores of the Mediterranean basin through their papers. In the first paper, Uri Dadush and Yana Myachenkova explain how trade agreements that the European Union has with North African countries are often seen as having delivered disappointing results, and how the agreements have been judged too harshly, as they helped generate large amounts of trade. In this regard, the paper gives relevant recommendations of policies for a greater trade performance. The second paper highlights the role of structural economic transformation as a necessary gateway for nations wishing to move up along the development path. It focuses on the case of Morocco, shedding light on its manufacturing sector\u2019s transformation, its integration in Global Value Chains (GVC) and its economic challenges. In \u201cThe EU-Southern Mediterranean Energy Relationship: A Fresh Perspective\u201d, Simone Tagliapietra analyses how regional energy cooperation should strongly focus on fostering large-scale deployment of renewable energy, allowing southern Mediterranean countries to meet their increasing energy demand in a more sustainable way, and having positive economic and political benefits for Europe

Singapore's Perspectives on the Proliferation of RTAs in East Asia and Beyond

In this article the viewpoints and adopted strategies of Singapore in managing its trading relationship with other countries in the world is discussed. Trade being the lifeline and basis for its economic survival, Singapore has been very active in the negotiation and establishment of Free Trade Agreements (FTAs) with countries, practically in every continent. Singapore's activism in bilateral FTAs is a reinforcement of its development strategy of expanding and ensuring market access for local and foreign investors. This is also in alignment with Singapore's pro- business strategy philosophy and the objective of ensuring economic viability. Also Singapore's trade accords go beyond trade liberalization and include “behind the border” impediments to trade; this is reckoned to keep the momentum of trade talk going and to hasten the process of liberalization by inducing other countries to liberalize.Free Trade Agreements, East Asia regionalism, trade liberalization, multilateralism,