By Hallard T. Croft, Kenneth Falconer, Richard K. Guy
Mathematicians and non-mathematicians alike have lengthy been enthusiastic about geometrical difficulties, fairly those who are intuitive within the experience of being effortless to nation, might be simply by an easy diagram. every one part within the ebook describes an issue or a gaggle of similar difficulties. frequently the issues are able to generalization of edition in lots of instructions. The booklet might be liked at many degrees and is meant for everybody from amateurs to investigate mathematicians.
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Extra resources for Unsolved Problems in Geometry: Unsolved Problems in Intuitive Mathematics
Pogorelov, On the determination of Fermi surfaces and electron velocities in metals by the oscillation of magnetic susceptibility, Dokl. Akad. N auk SSSR 96 (1954) 1143-1145. A. R. Mackintosh, The Fermi surface of metals, Scientific American, 209, no. 1. (July 1963), 110-120. D. Shoenberg, The de Haas-von Alphen effect, in The Fermi Surface, W. A. Harrison and M. B. ), Wiley, New York, 1960,74-83. J. Zaks, Nonspherical bodies with constant HA measurements exist, Amer. Math. Monthly 78 (1971) 513-516.
83 (1921) 311-319. A19. Extremal problems for elongated sets. Bieri poses the following problem for sets constrained to be "long and thin": Given the diameter and width of a plane convex set, what is the minimal possible perimeter length? It is conjectured that the extremal sets are polygons inscribed in a Reuleaux triangle. The other extremal problems of this type (maximum perimeter, minimum and maximum area) all seem to be susceptible to routine methods. Bieri further asks for a description of the plane convex sets of given diameter, width and perimeter that have maximal, and also that have minimal, area.
Let K be a ddimensional set of constant width 1. Let M be the collection of midpoints of the diametral chords of K, that is, the chords of K of length 1. What is the largest possible value of the diameter of M, taken over all such K? Can one always find ad-dimensional simplex containing M and contained in K? This last question is likely to be hard since an affirmative answer would imply the truth of Borsuk's conjecture (see Section 014). Z. A. Melzak, Problems connected with convexity, Problem 19, Canad.
Unsolved Problems in Geometry: Unsolved Problems in Intuitive Mathematics by Hallard T. Croft, Kenneth Falconer, Richard K. Guy