Unsolved Problems in Geometry

Notation and Definitions.- Sets. 1 Geometrical transformations..- Length, Area, and volume..- A. Convexity.- Al. The equichordal point problem..- A2. Hammer's x-ray problems..- A3. Concurrent normals..- A4. Billiard ball trajectories in convex regions..- A5. Illumination problems..- A6. The floating body problem..- A7. Division of convex bodies by lines or planes through a point..- A8. Sections through the centroid of a convex body..- A9. Sections of centro-symmetric convex bodies..- A10. What can you tell about a convex body from its shadows?.- A11. What can you tell About a convex body from its sections?.- A12. Overlapping convex bodies..- A13. Intersections of congruent surfaces..- A14. Rotating polyhedra..- A15. Inscribed and circumscribed centro-symmetric bodies..- A16. Inscribed affine copies of convex bodies..- A17. Isoperimetric inequalities and extremal problems..- A18. Volume against width..- A19. Extremal problems for elongated sets..- A20. Dido's problem..- A21. Blaschke's problem..- A22. Minimal bodies of constant width..- A23. Constrained is operimetric problems..- A24. Is a body Fairly round if all its sections are?.- A25. How far apart can various centers be?.- A26. Dividing up a piece of land by a short fence..- A27. Midpoints of diameters of sets of constant width..- A28. Largest convex hull of an arc of a given length..- A29. Roads on planets..- A30. The shortest curve cutting all the lines through a disk..- A31. Cones based on convex sets..- A32. Generalized ellipses..- A33. Conic sections through five points..- A34. The shape of worn stones..- A35. Geodesics..- A36. Convex sets with universal sections..- A37. Convex space-filling curves..- A38. m-convex sets..- B. Polygons, Polyhedra, and Polytopes.- Bl. Fitting one triangle inside another..- B2. Inscribing polygons in curves..- B3. Maximal regular polyhedra inscribed in regular polyhedra..- B4. Prince Rupert's problem..- B5. Random polygons and polyhedra..- B6. Extremal problems for polygons..- B7. Longest chords of polygons..- B8. Isoperimetric inequalities for polyhedra..- B9. Inequalities for sums of edge lengths of polyhedra..- B10. Shadows of polyhedra..- B11. Dihedral angles of polyhedra..- B12. Monostatic polyhedra..- B13. Rigidity of polyhedra..- B14. Rigidity of frameworks..- B15. Counting polyhedra..- B16. The sizes of the faces of a polyhedron..- B17. Unimodality of f-vectors of polytopes..- B18. Inscribable and circumscribable polyhedra..- B19. Truncating polyhedra..- B20. Lengths of paths on polyhedra..- B21. Nets of polyhedra..- B22. Polyhedra with congruent faces..- B23. Ordering the faces of a polyhedron..- B24. The four color conjecture for toroidal polyhedra..- B25. Sequences of polygons and polyhedra..- C. Tiling and Dissection.- Cl. Conway's fried potato problem..- C2. Squaring the square..- C3. Mrs. Perkins's quilt..- C4. Decomposing a square or a cube into n smaller ones..- C5. Tiling with incomparable rectangles and cuboids..- C6. Cutting up squares, circles, and polygons..- C7. Dissecting a polygon into nearly equilateral triangles..- C8. Dissecting the sphere into small congruent pieces..- C9. The simplexity of the d-cube..- C10. Tiling the plane with squares..- C11. Tiling the plane with triangles..- C12. Rotational symmetries of tiles..- C13. Tilings with a constant number of neighbors..- C14. Which polygons tile the plane?.- C15. Isoperimetric problems for tilings..- C16. Polyominoes..- C17. Reptiles..- C18. Aperiodic tilings..- C19. Decomposing a sphere into circular arcs..- C20. Problems in equidecomposability..- D. Packing and Covering.- D1. Packing circles, or spreading points, in a square..- D2. Spreading points in a circle..- D3. Covering a circle with equal disks..- D4. Packing equal squares in a square..- D5. Packing unequal rectangles and squares in a square..- D6. The Rados' problem on selecting disjoint squares..- D7. The problem of Tammes..- D8. Covering the sphere with circular caps..- D9. Variations on the penny-packing problem..- D10. Packing Balls in space..- D11. Packing and covering with congruent convex sets..- D12. Kissing numbers of convex sets..- D13. Variations on Bang's plank theorem..- D14. Borsuk's conjecture..- D15. Universal covers..- D16. Universal covers for several sets..- D17. Hadwiger's covering conjecture..- D18. The worm problem..- E. Combinatorial Geometry.- El. Helly-type problems..- E2. Variations on Krasnosel'skii's theorem..- E3. Common transversals..- E4. Variations on Radon's theorem..- E5. Collections of disks with no three in a line..- E6. Moving disks around..- E7. Neighborly convex bodies..- E8. Separating objects..- E9. Lattice point problems..- E10. Sets covering constant numbers of lattice points..- Ell. Sets that can be moved to cover several lattice points..- E12. Sets that always cover several lattice points..- E13. Variations on Minkowski's theorem..- E14. Positioning convex sets relative to discrete sets..- F. Finite Sets of Points.- F1. Minimum number of distinct distances..- F2. Repeated distances..- F3. Two-distance sets..- F4.Can each distance occur a different number of times?.- F5. Well-spaced sets of points..- F6. Isosceles triangles determined by a set of points..- F7. Areas of triangles determined by a set of points..- F8. Convex polygons determined by a set of points..- F9. Circles through point sets..- F10. Perpendicular Bisectors..- F11. Sets cut off by straight lines..- F12. Lines through sets of points..- F13. Angles determined by a set of points..- F14. Further problems in discrete geometry..- F15. The shortest path joining a set of points..- F16. Connecting points by arcs..- F17. Arranging points on a sphere..- G. General Geometric Problems.- G1. Magic numbers..- G2. Metrically homogeneous sets..- G3. Arcs with increasing chords..- G4. Maximal sets avoiding certain distance configurations..- G5. Moving furniture around..- G6. Questions related to the Kakeya problem..- G7. Measurable sets and lines..- G8. Determining curves from intersections with lines..- G9. Two sets which always intersect in a point..- G10. The chromatic number of the plane and of space..- G11. Geometric graphs..- G12. Euclidean Ramsey problems..- G13. Triangles with vertices in sets of a given area..- G14. Sets Containing large triangles..- G15. Similar copies of sequences..- G16. Unions of similar copies of sets..- Index of Authors Cited.- General Index.