Converses of Gauss’ theorem on the arithmetic mean

Some Converses of the Strong Separation Theorem

A convex subset B of a real locally convex space X is said to have the separation property if it can be separated from any closed convex subset A of X, which is disjoint from B, by a closed hyperplane. The strong separation theorem says that if B is weakly compact then it has the separation property. In this paper, we present several versions for the converse and discuss some applications. For ...

Sharp Two Parameter Bounds for the Logarithmic Mean and the Arithmetic–geometric Mean of Gauss

For fixed s 1 and t1,t2 ∈ (0,1/2) we prove that the inequalities G(t1a + (1− t1)b,t1b+(1− t1)a)A1−s(a,b) > AG(a,b) and G(t2a+(1− t2)b,t2b+(1− t2)a)A1−s(a,b) > L(a,b) hold for all a,b > 0 with a = b if and only if t1 1/2− √ 2s/(4s) and t2 1/2− √ 6s/(6s) . Here G(a,b) , L(a,b) , A(a,b) and AG(a,b) are the geometric, logarithmic, arithmetic and arithmetic-geometric means of a and b , respectively....

On Converses of Napoleon’s Theorem and a Modified Shape Function

The (negative) Torricelli triangle T1(ABC) of a nondegenerate (positively oriented) triangle ABC is defined to be the triangle A1B1C1, where ABC1, BCA1, and CAB1 are the equilateral triangles drawn outwardly on the sides of ABC. It is known that not every triangle is the Torricelli triangle of some initial triangle, and triangles that are not Torricelli triangles are characterized in [28]. In t...

The Gauss-Bonnet Theorem

Remarks on the Gauss-Green Theorem

These notes cover material related to the Gauss-Green theorem that was developed for work with S. Hofmann and M. Mitrea, which appeared in [HMT].