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.

We prove that there are no bounded domains with smooth boundaries in even-dimensional Euclidean spaces, such that the volumes cut off from them by affine hyperplanes depend algebraically on these hyperplanes. For convex ovals in R2, this is the Newton’s Lemma XXVIII, see [11], [14], [2], [3].

Article history: Received 4 August 2011 Available online 11 November 2013

Let {(Zi,Wi) : i = 1, . . . , n} be uniformly distributed in [0, 1]d × G(k, d), where G(k, d) denotes the space of k-dimensional linear subspaces of Rd. For a differentiable function f : [0, 1]k → [0, 1]d, we say that f interpolates (z,w) ∈ [0, 1]d × G(k, d) if there exists x ∈ [0, 1]k such that f(x) = z and ~ f(x) = w, where ~ f(x) denotes the tangent space at x defined by f . For a smoothness...

Locality-sensitive hashing converts high-dimensional feature vectors, such as image and speech, into bit arrays and allows high-speed similarity calculation with the Hamming distance. There is a hashing scheme that maps feature vectors to bit arrays depending on the signs of the inner products between feature vectors and the normal vectors of hyperplanes placed in the feature space. This hashin...

We consider the quantum hyperplane x ' x J = q , j x J x ' (i, j = 1..n) and define and consider = 'J and fl'J are complex numbers. We deformations of the form x ' x J q,~ x J x ~ + Zk ~ J X k + fl'J, where ~k prove that for generic q,j no nontrivial deformations exist for n/> 3 Mathematics Subject Classifications (1991). 17B37, 16E40.