Level of Sets on Spheres

Citation

Sonneborn, Lee Myers (1956) Level of Sets on Spheres. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/rjtw-fw15. https://resolver.caltech.edu/CaltechTHESIS:07012025-154007671

Abstract

Let f:Sⁿ X I¹ → E¹ be a continuous, real-valued function on Sⁿ X I¹ for > 1. Then for every t Ɛ I¹ there is a subset A_t X t of the n-sphere Sⁿ X t with the following properties:

i) f(A_t X t) = k_t independent of x Ɛ A_t.

ii) A_t X t is connected.

iii) (Sⁿ X t) - (A_t X t) has no component containing more than half the n-dimensional measure of Sⁿ X t.

iv) For any measure-preserving homeomorphism, g, of Sⁿ X t, A_t X t contains the image of at least one of its points. (e.g. A_t X t contains a pair of antipodal points of Sⁿ X t)

v) k_t varies continuously with t.

Further, if g:T² E¹ is a continuous real-valued function defined on a torus, then there is a connected, non-contractible subset of T²on on which g is constant.

Thesis Files