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We give two new proofs of the fundamental triangle inequality. By using the fundamental triangle inequality and the related Blundon’s theorem, we prove several new general results. We also obtain two new generalizations of Gerretsen’s double inequality, from which some new and old refinements of Gerretsen’s double inequality are obtained as well. In addition, we propose a conjecture related to the fundamental triangle inequality. [ABSTRACT FROM AUTHOR]

We improve Euler's inequality R ≥ 2 r where R and r are triangle's circumradius and inradius, respectively. It involves symmetric functions of triangle's sidelengths. We also prove non-Euclidean versions of this result. Next, we refine 3D analogue of Euler's inequality R ≥ 3 r for tetrahedra and briefly discuss recursive way to improve Euler's inequality for simplices. [ABSTRACT FROM AUTHOR]

We establish a new generalization of the Erdös–Mordell inequality by adding more a set of weights to its terms. The same method is used on two other variants of the Erdös–Mordell inequality which are Barrow's inequality and Dao–Nguyen–Pham's inequality. Using these generalizations, we derived some strengthened versions of the original Erdös–Mordell inequality and its variations. [ABSTRACT FROM AUTHOR]

We investigate the strict power concavity of the convolution of a function and the characteristic function of a convex body. Using the result, we show that every convex body has a unique illuminating center. [ABSTRACT FROM AUTHOR]

Given a triangle A A A and weights m, m, m, a geometric proof is given of (a) the existence of a point F whose weighted distance sum has the smallest possible value, (b) the value of this minimum, and (c) the construction of the point. A characterization of the weights that guarantees the point is inside the triangle is given. This is then used to supply solutions to three extremal problems, as well as to generate sharp inequalities connecting the elements of triangle A A A. [ABSTRACT FROM AUTHOR]

We derive several results in classical Euclidean elementary geometry using the steering ellipsoid formalism from quantum mechanics. This gives a physically motivated derivation of very non-trivial geometric results, some of which are entirely new. We consider a sphere of radius r contained inside another sphere of radius R, with the sphere centres separated by distance d. When does there exist a 'nested' tetrahedron circumscribed about the smaller sphere and inscribed in the larger? We derive the Grace-Danielsson inequality $${d^{2}\leq (R+r)(R-3r)}$$ as the sole necessary and sufficient condition for the existence of a nested tetrahedron. Our method also gives the condition $${d^{2}\leq R(R-2r)}$$ for the existence of a nested triangle in the analogous two-dimensional scenario. These results imply the Euler inequality in two and three dimensions. Furthermore, we formulate a new inequality that applies to the more general case of ellipses and ellipsoids. [ABSTRACT FROM AUTHOR]

In the paper, by using the important 'R-r-s' method in the theory of triangle inequalities and employing mathematical software Maple for the calculations, the author refines an equivalent form of one of the Gerretsen inequalities. We also propose some interesting conjectures which have been checked by computer. [ABSTRACT FROM AUTHOR]