An ancient optics problem of Ptolemy, studied later by Alhazen, is discussed. This problem deals with reflection of light in spherical mirrors. Mathematically this reduces to the solution of a quartic equation, which we solve and analyze using a symbolic computation software. Similar problems have been recently studied in connection with ray-tracing, catadioptric optics, scattering of electromagnetic waves, and mathematical billiards, but we were led to this problem in our study of the so-called triangular ratio metric.2010 Mathematics Subject Classification. 30C20, 30C15, 51M99.

Answering a question about triangle inequality suggested by R. Li, A. Barrlund [3] introduced a distance function which is a metric on a subdomain of R n . We study this Barrlund metric and give sharp bounds for it in terms of other metrics of current interest. We also prove sharp distortion results for the Barrlund metric under quasiconformal maps.2010 Mathematics Subject Classification. 30C20, 30C15, 51M99.

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To cite this article: Marcelina Mocanu (2010) A generalization of Orlicz-Sobolev spaces on metric measure spaces via Banach function spaces, Complex Variables andWe show that several basic definitions and results regarding modulus, capacity and Orlicz-Sobolev spaces on metric measure spaces can be generalized to the case where the role of the Orlicz space is played by an abstract Banach function space B. This new general setting could bring a new perspective in the study of Sobolev-type spaces on metric measure spaces, due to the great generality of Banach function spaces. We prove several properties of the newly introduced Sobolev-type space N 1,B (X ), including its completeness and a Mazur-type theorem.

Abstract. In this paper we deal with a metric measure space equipped with a doubling measure and supporting an Orlicz-Poincaré inequality, namely a weak (1, Φ)-Poincaré inequality, that is more general than the (1, 1)-Poincaré inequality. For a wide class of Orlicz spaces, we prove that the corresponding Orlicz-Sobolev functions have Lebesgue points outside a set of zero Orlicz-Sobolev capacity. This results extends a theorem of Tuominen (2009)