Illumination problem and absolutely focusing mirrors

We consider the illumination and the strong illumination properties for closed bounded regions of Euclidean spaces. These properties are intimately connected with a problem of chaoticity of the corresponding billiards. It is shown that there are only two mechanisms of chaoticity in billiard systems, which are called the mechanism of dispersing and the mechanism of defocusing. Our results show how the regions with different illumination properties should be designed. Especially each focusing mirror in the boundary of a region must be an absolutely focusing one. The notion of absolutely focusing mirrors is a new one in the geometric optic and it plays a key role for the illumination problem.

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