Master's Thesis

What is the minimum number of light sources needed to illuminate a d-dimensional convex body? This question, with its conjectured answer of 2d, is known today as the illumination conjecture. It dates back to 1960 and is due to V. Boltyanski [1] and H. Hadwiger [3]. While still open in dimensions three and greater, some specific cases of the illumination conjecture have been verified. For instance, B.V. Dekster proved the illumination conjecture for three-dimensional, plane symmetric, convex bodies [2].

The main objective of my thesis was to give a rigorous exposition of Dekster's proof. I was able to develop constructive proofs for a number of cases including the cases where the base set: has no sides with the help of Mazur's finite dimensional density theorem; has an odd number of sides; has a degenerate side; and has a side of length less than 1/2 after applying a certain linear transformation and then using the John-Löwner theorem.

References

  1. V.G. Boltyanski. The problem of illuminating the boundary of a convex body. Izv. Mold. Filiala Akad. Nauk SSSR, 10(76):77-84, 1960.
  2. Boris Dekster. Each convex body in E3 symmetric about a plane can be illuminated by 8 directions. Journal of Geometry, 69(1):37-50, 2000.
  3. H. Hadwiger. Ungelöste Probleme. Elemente der Mathematik, 15(6):130-131, 1960.

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