Recent Advances in Geometric Inequalities

The geometric proof to a sharp version of Blundon's inequalities

“…If φ = min{|A − B|,|B −C|,|C − A|}, then we proved the inequality − cos φ cos ION cos φ , where O is the circumcenter, I is the incenter, and N is the Nagel point of triangle ABC . As a direct consequence, we obtain a sharp version to Gerretsen's inequalities [7].…”

Generalized power means and interpolating inequalities

“…In Ivan Niven's book Maxima and Minima Without Calculus [14] that has enlightened many interesting articles in MAA journals, inequalities involving different means are key ingredients. In fact, geometric inequalities which are derived from inequalities of means are collected in many books such as [6,13,14,16,17] etc. Regarding a = (a 1 , a 2 , · · · , a n ) as the side lengths of an n-sided plane polygon P n , in [7], we have used some refined inequalities of means to establish many geometric inequalities of isoperimetric type for polygons.…”

“…Regarding a = (a 1 , a 2 , · · · , a n ) as the side lengths of an n-sided plane polygon P n , in [7], we have used some refined inequalities of means to establish many geometric inequalities of isoperimetric type for polygons. In [13], one can find many geometric inequalities for an n-simplex S n in R m where a = (a 1 , a 2 , · · · , a n ) represents a collection of the areas of its n faces, or a collection of its n altitudes, and so on, as immediate consequences of some inequalities in means. The method of interpolating inequalities is certainly applicable to refining geometric inequalities.…”

“…For example, the λ-method by D. S. Mitrinović and P. M. Vasić, and the functional equation approach by S. Iwamoto are discussed in [12]. Due to numerous references on inequalities and the large variety of methods, we shall recommend to the interested reader the excellent books [3,4,5,11,12,13] and further relevant articles therein.…”

Some inequalities for chord power integrals of parallelotopes

“…For alternative denitions, historical background and further details related with Schurconvexity the reader is referred to the monographs [7], [8], and [14].…”