An Isoperimetric Inequality in the Theory of Self-Contained Numerical Integration Formulas

摘要

A closed region R that satisfies the conditions the integral over R of ((x sub i) (x sub j)dx) = delta sub ij I sub zero, where delta sub ij is the Kronecker delta, is said to be Stroud normal. The C-radius rho(R) is the radius of the largest sphere centered at the centroid of R that is contained in R. Conjecture 1: Among all convex Stroud normal subsets of E superscript n7 rho assumes its minimum on the n-simplex and its maximum on the closed ball. It is shown how this conjecture arose in the theory of self-contained numerecal integration formulas. In the case n = 2 the conjecture is proved for regular polygons and is shown to be equivalent to Conjecture 2: Among all convex Stroud normal m-gons, p-rho assumes its minimum on the triangle and its maximum on the regular m-gon. The latter conjecture is proved for m = 4 and numerical evidence is presented in support of the conjecture for 5 = or