Approximation of the Euclidean ball by polytopes with a restricted number of facets

We prove that there is an absolute constant $C$ such that for every $n \geq 2$ and $N\geq 10^n,$ there exists a polytope $P_{n,N} \subset \mathbb{R}^n$ with at most $N$ facets that satisfies $$\Delta_{v}(D_n,P_{n,N}):=\text{vol}_n\left(D_n \Delta P_{n,N}\right)\leq Cn^{-2/(n-1}\text{vol}_n\left(D_n\right)$$ and $$\Delta_{s}(D_n,P_{n,N}):=\text{vol}_{n-1}\left(\partial\left(D_n\cup P_{n,N}\right)\right) - \text{vol}_{n-1}\left(\partial\left(D_n\cap P_{n,N}\right)\right) \leq 4CN^{-\frac{2}{n-1}} \text{vol}_{n-1}\left(\partial D_n\right),$$ where $D_n$ is the $n$-dimensional Euclidean unit ball. This result closes gaps from several papers of Hoehner, Ludwig, Sch\"utt and Werner. The upper bounds are optimal up to absolute constants. This result shows that a polytope with an exponential number of facets (in the dimension) can approximate the $n$-dimensional Euclidean ball with respect to the aforementioned distances

Secondary Liability for Trademark Infringement on the Internet: The Situation in Germany and Throughout the EU

It is a safe assumption that every one of the twenty-eight EU Member States has some form of secondary liability principles within its general tort law. Traditionally, tort law principles require an element of deliberate contribution to, and therefore positive knowledge of, another person’s action. They are therefore of limited value for situations like Internet infringement, because the contributor is typically unaware of persons committing concrete acts of infringement. When the contributor merely provides the technical infrastructure that is used by an anonymous crowd for legitimate as well as illegitimate purposes, the “deliberate contribution” or “positive knowledge” element is difficult to prove. The general principles of tort law are not harmonized within Europe. However, regarding Internet services that may be misused for unlawful acts (including, but not limited to, intellectual property infringement), the E-Commerce Directive has created a legal framework for exemptions from liability (the safe harbor). In that regard, the E-Commerce Directive is the European equivalent of the U.S. Digital Millennium Copyright Act (DMCA), with the notable distinction that the E- Commerce Directive applies horizontally, thus encompassing trademark infringement