Symmetry Points of a Convex Set: Basic Properties and Computational Complexity

SYMMETRY POINTS OF A CONVEX SET : Basic Properties and Computational Complexity

Given a convex body S ⊂ IRn and a point x ∈ S, let sym(x, S) denote the symmetry value of x in S: sym(x, S) := max{α ≥ 0 : x + α(x− y) ∈ S for every y ∈ S} , which essentially measures how symmetric S is about the point x, and define sym(S) := max x∈S sym(x, S) . We call x∗ a symmetry point of S if x∗ achieves the above supremum. These symmetry measures are all invariant under invertible affine...

Computational complexity of fixed points

A review of computational complexity results for approximating fixed points of Lipschitz functions is presented. Univariate and multivariate results are summarized for the second and infinity norm cases as well as the absolute, residual and relative error criteria. Contractive, nonexpansive, directionally nonexpansive, and expansive classes of functions are considered and optimal or nearly opti...

Basic Properties of Convex Polytopes

Convex polytopes are fundamental geometric objects that have been investigated since antiquity. The beauty of their theory is nowadays complemented by their importance for many other mathematical subjects, ranging from integration theory, algebraic topology, and algebraic geometry (toric varieties) to linear and combinatorial optimization. In this chapter we try to give a short introduction, pr...

Some Properties of Certain Subclasses of Close-to-Convex and Quasi-convex Functions with Respect to 2k-Symmetric Conjugate Points

The Computational Complexity ofRelating Points

Vilain has suggested the point-interval algebra for reasoning about the relative positions of points and intervals on the real line. The satissability problem for the full point-interval algebra is known to be NP-complete but the computational complexity of its 4; 294; 967; 296 subclasses has not been investigated. We classify these subclasses with respect to whether their corresponding satissa...