Symplectic Banach-Mazur distances between subsets of C^n

Following proposals of Ostrover and Polterovich, we introduce and study "coarse" and "fine" versions of a symplectic Banach-Mazur distance on certain open subsets of $\mathbb{C}^n$ and other open Liouville domains. The coarse version declares two such domains to be close to each other if each domain admits a Liouville embedding into a slight dilate of the other; the fine version, which is similar to the distance on subsets of cotangent bundles of surfaces recently studied by Stojisavljevi\'c and Zhang, imposes an additional requirement on the images of these embeddings that is motivated by the definition of the classical Banach-Mazur distance on convex bodies. Our first main result is that the coarse and fine distances are quite different from each other, in that there are sequences that converge coarsely to an ellipsoid but diverge to infinity with respect to the fine distance. Our other main result is that, with respect to the fine distance, the space of star-shaped domains in $\mathbb{C}^n$ admits quasi-isometric embeddings of $\mathbb{R}^D$ for every finite dimension $D$. Our constructions are obtained from a general method of constructing $(2n+2)$-dimensional Liouville domains whose boundaries have Reeb dynamics determined by certain autonomous Hamiltonian flows on a given $2n$-dimensional Liouville domain. The bounds underlying our main results are proven using filtered equivariant symplectic homology via methods from prior joint work with Gutt.Comment: v2: 56 pages, several corrections and additional explanations based on referee's comments. To appear in J. Top. Ana

Three Stories about the Chance of Casting a Pivotal Vote

People vote from self-interest or from a sense of duty. Voting from self-interest requires there to be some chance, however small, that one's vote swings the outcome of the election from the political party one opposes to the political party one favours. This paper is a discussion of three models of how that chance might arise: the common sense model inferring the probability of a tied vote today from the distribution of outcomes in past elections, person-to-person randomization where each voter looks upon the political preferences of rest of the electorate as analogous to drawings from an urn with given proportions of red and blue balls, and nation-wide randomization where voters are lined up according to their valuations (positive or negative) of a win for one of the two competing parties, but where chance shifts the entire schedule of preferences up or down. Emphasis is on the third model about which this paper may have something new to say.Pivotal voting, Duty to vote, Compulsory voting

The Marginal Cost of Public Funds is the Ratio of Mean Income to median Income

The marginal cost of public funds is the equilibrium price at the intersection of the appropriately-defined demand curve for and the supply curve of public expenditure. In a world with identical people and with no excess burden of taxation, that price would have to be 1. Otherwise the median voter's choice of a demogrant - or of its opposite, a head tax - fixes the marginal cost of public funds at the ratio of the mean income to the median income. A proof of this assertion is presented not for its realism, but because it calls attention to the interaction of the different influences upon the marginal cost of public funds.Marginal Cost of Public Funds

The Distributive Implications of Patents on Indivisible Goods

Patents raise the price and reduce consumption of the patented good, but the resulting deadweight loss is thought to be worth bearing when patent protection is required as an incentive to invention. The newly-invented good generates a residual surplus, making people better off than they would be if the good had not been invented. This well-known argument is usually framed in a context where people are identical, everybody's demand curve for the newly-invented good is the same and everybody shares to some extent in the residual surplus. However, when the newly-invented good is indivisible - like a heart transplant or the treatment of AIDS, where, in effect, a person consumes either one full unit of the good or none - the effect of a patent is to concentrate the entire benefit of the patented good upon the rich, leaving the poor no better off than if the good had not been invented.Patents, Indivisible Goods