Steiner symmetrization using a finite set of directions

Let $v_1, ..., v_m$ be a finite set of unit vectors in \RR^n. Suppose that an infinite sequence of Steiner symmetrizations are applied to a compact convex set $K$ in \RR^n, where each of the symmetrizations is taken with respect to a direction from among the $v_i$. Then the resulting sequence of Steiner symmetrals always converges, and the limiting body is symmetric under reflection in any of the directions $v_i$ that appear infinitely often in the sequence. In particular, an infinite periodic sequence of Steiner symmetrizations always converges, and the set functional determined by this infinite process is always idempotent.Comment: 18 pages. The essential results are the same as in the previous version. This version includes a more thorough introduction, some clarifications in the proofs, an updated bibliography, and some open questions at the en

Bonnesen-type inequalities for surfaces of constant curvature

A Bonnesen-type inequality is a sharp isoperimetric inequality that includes an error estimate in terms of inscribed and circumscribed regions. A kinematic technique is used to prove a Bonnesen-type inequality for the Euclidean sphere (having constant positive Gauss curvature) and the hyperbolic plane (having constant negative Gauss curvature). These generalized inequalities each converge to the classical Bonnesen-type inequality for the Euclidean plane as the curvature approaches zero.Comment: Latex, 11 page

Impact of technological synchronicity on prospects for CETI

For over 50 years, astronomers have searched the skies for evidence of electromagnetic signals from extraterrestrial civilizations that have reached or surpassed our level of technological development. Although often overlooked or given as granted, the parallel use of an equivalent communication technology is a necessary prerequisite for establishing contact in both leakage and deliberate messaging strategies. Civilization advancements, especially accelerating change and exponential growth, lessen the perspective for a simultaneous technological status of civilizations thus putting hard constraints on the likelihood of a dialogue. In this paper we consider the mathematical probability of technological synchronicity of our own and a number of other hypothetical extraterrestrial civilizations and explore the most likely scenarios for their concurrency. If SETI projects rely on a fortuitous detection of leaked interstellar signals (so called "eavesdropping") then without any prior assumptions N \geq 138-4991 Earth-like civilizations have to exist at this moment in the Galaxy for the technological usage synchronicity probability p \geq 0.95 in the next 20 years. We also show that since the emergence of complex life, coherent with the hypothesis of the Galactic habitable zone, N \geq 1497 extraterrestrial civilizations had to be created in the Galaxy in order to achieve the same estimated probability in the technological possession synchronicity which corresponds to the deliberate signaling scenario.Comment: 15 pages, 7 figures, 1 table, accepted for publication in the International Journal of Astrobiolog

Ecologically sustainable but unjust? Negotiating equity and authority in common-pool marine resource management

Under appropriate conditions, community-based fisheries management can support sound resource stewardship, with positive social and environmental outcomes. Evaluating indigenous peoples’ involvement in commercial sea cucumber and geoduck fisheries on the central coast of British Columbia, Canada, we found that the current social-ecological system configuration is relatively ecologically sustainable according to stock assessments. However, the current system also results in perceived inequities in decision-making processes, harvesting allocations, and socioeconomic benefits. As a result, local coastal resource managers envision a transformation of sea cucumber and geoduck fisheries governance and management institutions. We assessed the potential robustness of the proposed institutions using Elinor Ostrom’s common-pool resource design principles. Grounded in the region’s legal, political, and historical context, our analysis suggests that greater local involvement in these invertebrate fisheries and their management could provide more benefits to local communities than the status quo while maintaining an ecologically sustainable resource. Our research highlights the importance of explicitly addressing historical context and equity considerations in social-ecological system analyses and when renegotiating the institutions governing common-pool resources