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.

In the present paper, we first prove the logarithmic convexity of the elementary function b x −a x x , where x 6= 0 and b > a > 0. Basing on this, we then provide a simple proof for Schur-convex properties of the extended mean values, and, finally, discover some convexity related to the extended mean values.

Khrapchenko’s classical lower bound n2 on the formula size of the parity function f can be interpreted as designing a suitable measure of sub-rectangles 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(n2) and show that several measures considered fo...