We characterize C embedded hypersurfaces of R as the only locally closed sets with continuously varying flat tangent cones whose measuretheoretic-multiplicity is at most m < 3/2. It follows then that any (topological) hypersurface which has flat tangent cones and is supported everywhere by balls of uniform radius is C. In the real analytic case the same conclusion holds under the weakened hypot...

The familiar bijections between the representations of permutations as words and as products of cycles have a natural class of “data driven” extensions that permit us to use purely combinatorial means to obtain precise probabilistic information about the geometry of random walks. In particular, we show that the algorithmic bijection of Bohnenblust and Spitzer can be used to obtain means, varian...

Khrapchenko’s classical lower bound n on the formula size of the parity function f can be interpreted as designing a suitable measure of subrectangles of the combinatorial rectangle f−1(0)× f−1(1). Trying to generalize this approach we arrived at the concept of convex measures. We prove the negative result that convex measures are bounded by O(n) and show that several measures considered for pr...

The problem of minimizing f̃ = f +p over some convex subset of a Euclidean space is investigated, where f(x) = x Ax + b x is a strictly convex quadratic function and |p| is only assumed to be bounded by some positive number s. It is shown that the function f̃ is strictly outer γ-convex for any γ > γ∗, where γ∗ is determined by s and the smallest eigenvalue of A. As consequence, a γ∗-local minimal...

As a generalization of geodesic function, this paper introduces the notion φE-convex function. Some properties function and are established. The concepts set φE-epigraph also given. characterization functions in terms their φE-epigraphs, obtained.