In this note, we prove that smooth self-shrinkers in R, that are entire graphs, are hyperplanes. Previously Ecker and Huisken showed that smooth self-shrinkers, that are entire graphs and have at most polynomial growth, are hyperplanes. The point of this note is that no growth assumption at infinity is needed.

The aim of this paper is to provide the sufficient condition for a mass distribution in R to admit an equipartition with a collection of hyperplanes some of which are parallel. The results extend the previously obtained results for the equipartitions with non-parallel hyperplanes. (See [4].)

Let A be a Coxeter hyperplane arrangement, that is the arrangement of reflecting hyperplanes of an irreducible finite Coxeter group. A deformation of A is an affine arrangement each of whose hyperplanes is parallel to some hyperplane of A. We survey some of the interesting combinatorics of classes of such arrangements, reflected in their characteristic polynomials.

In 1982–83, E. Nochka proved a conjecture of Cartan on defects of holomorphic curves in P relative to a possibly degenerate set of hyperplanes. This was further explained by W. Chen in his 1987 thesis, and subseqently simplified by M. Ru and P.-M. Wong in 1991. The proof involved assigning weights to the hyperplanes. This paper provides a mild simplification of the proof of the construction of ...

Given a symmetric convex body C and n hyperplanes in an Euclidean space, there is a translate of a multiple of C, at least 1 n+1 times as large, inside C, whose interior does not meet any of the hyperplanes. The result generalizes Bang's solution of the plank problem of Tarski and has applications to Diophantine approximation.