St. Petersburg Department, Steklov Mathematical Institute, St. Petersburg, Russia. e-mail: zalg@wisdom.weizmann.ac.il

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Abstract

What is the form of the shortest curve C going outside the unit sphere S in ℝ3 such that passing along C we can see all points of S from outside? How will the form of C change if we require that C has one (or both) of its endpoints on S? A solution to the latter problem also answers the following question. You are in a half-space at a unit distance from the boundary plane P, but you do not know where P is. What is the shortest space curve C such that going along C you will certainly come to P? Geometric arguments suggest that the required curves should be looked for in certain classes depending on several parameters. A computer-aided analysis yields the best curves in the classes. Some other questions are solved in a similar way. Bibliography: 4 titles.