Vladimir Volenec

Recent advances in geometric inequalities

The Existence of a Triangle.- Duality between Geometric Inequalities and Inequalities for Positive Numbers.- Homogeneous Symmetric Polynomial Geometric Inequalities.- Duality between Different Triangle Inequalities and Triangle Inequalities with (R, r, s).- Transformations for the Angles of a Triangle.- Some Trigonometric Inequalities.- Some Other Transformations.- Convex Functions and Geometric Inequalities.- Miscellaneous Inequalities with Elements of a Triangle.- Special Triangles.- Triangle and Point.- Inequalities with Several Triangles.- The Mobius-Neuberg and the Mobius-Pompeiu Theorems.- Inequalities for Quadrilaterals.- Inequalities for Polygons.- Inequalities for a Circle.- Particular Inequalities in Plane Geometry.- Inequalities for Simplexes in En (n ? 2).- Inequalities for Tetrahedra.- Other Inequalities in En (n ? 2).

Bounds for the Differences of Matrix Means

The classical inequality between the weighted arithmetic and geometric means has recently been extended to means of positive definite matrices. Here we give bounds for the difference between such matrix means. Differences between other matrix means, such as the geometric and harmonic means, are also given. Finally, conditions for the reversal of the matrix inequalities obtained are also pointed out.

THE ARITHMETIC-GEOMETRIC-HARMONIC-MEAN AND RELATED MATRIX INEQUALITIES

Abstract Recently, Sagae and Tanabe defined a geometric mean of positive definite matrices and proved the harmonic-geometric-arithmetic-mean inequality. Here, we give a reversal of these results.

Some Trigonometric Inequalities

In the book [1] one finds that almost all triangles inequalities are symmetric in form when expressed in terms of the sides a, b, c or the angles A, B, C of a given triangle. No doubt that also assymmetric triangle inequalities play a very important role in geometric inequalities. It should be noted that many of these inequalities are still valid for real numbers A, B, C which satisfy the condition

Thébault's theorem

A + B + C = p\pi ,

Inequalities for Volumes of Simplices in Terms of Their Faces

where p is a natural number (which has to be odd in some cases). This also applies to the inequality of M. S. Klamkin [2] which can be specialized in many ways to obtain numerous well known inequalities.

The Existence of a Triangle

It will be shown that the affine fullerene C60, which is defined as an affine image of buckminsterfullerene C60, can be obtained only by means of the golden section. The concept of the affine fullerene C60 will be constructed in a general GS-quasigroup using the statements about the relationships between affine regular pentagons and affine regular hexagons. The geometrical interpretation of all discovered relations in a general GS-quasigroup will be given in the GS-quasigroup .

The Möbius-Neuberg and the Möbius-Pompeiu Theorems

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