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From a fixed acute triangular base Δ A B C , all possible tetrahedra in three-dimensional real space are considered. The possible angles at the additional vertex P are shown to be bounded by certain inequalities, mostly linear inequalities. Together, these inequalities provide fairly tight bounds on the possible angle combinations at P. Four sets of inequalities are used for this purpose, though the inequalities in the first set are rather trivial. The inequalities in the second set can be established quickly, but do not seem to be known. The third and fourth set of inequalities are proved by studying scalar and vector fields on toroids. The first three sets of inequalities are linear in the angles at P, but the last set involves cosines of these angles. A generalization of the last two sets of inequalities is also proved, using the Poincaré–Hopf Theorem. Extensive testing of these results has been done using Mathematica and C++. The C++ code for this is listed in an appendix. While it has been demonstrated that the inequalities bound the possible combinations of angles at P, the results also reveal that additional inequalities, in particular linear inequalities, exist that would provided tighter bounds. [ABSTRACT FROM AUTHOR]

In this paper, we investigate an extremum problem for the power moment of a convex polygon contained in a disc. Our result is a generalization of a classical theorem: among all convex n-gons contained in a given disc, the regular n-gon inscribed in the circle (up to rotation) uniquely maximizes the area functional. It also implies that, among all convex n-gons contained in a given disc and containing the center in those interiors, the regular n-gon inscribed in the circle (up to rotation) uniquely maximizes the mean of the length of the chords passing through the center of the disc. [ABSTRACT FROM AUTHOR]

Inequalities for the lengths of edges in trihedrals of the eep( I) and hhp( I) types on a hyperbolic plane $$\widehat{H}$$ of positive curvature are proved. Based on these inequalities, the extremal property of elliptic and hyperbolic segments and the extremal property of the diagonals in a simple 4-contour on the plane $$\widehat{H}$$ are obtained. [ABSTRACT FROM AUTHOR]

This paper studies four families of polyhedral norms parametrized by a single parameter. The first two families consist of the CVaR norm (which is equivalent to the D-norm, or the largest- $$k$$ norm) and its dual norm, while the second two families consist of the convex combination of the $$\ell _1$$ - and $$\ell _\infty $$ -norms, referred to as the deltoidal norm, and its dual norm. These families contain the $$\ell _1$$ - and $$\ell _\infty $$ -norms as special limiting cases. These norms can be represented using linear programming (LP) and the size of LP formulations is independent of the norm parameters. The purpose of this paper is to establish a relation of the considered LP-representable norms to the $$\ell _p$$ -norm and to demonstrate their potential in optimization. On the basis of the ratio of the tight lower and upper bounds of the ratio of two norms, we show that in each dual pair, the primal and dual norms can equivalently well approximate the $$\ell _p$$ - and $$\ell _q$$ -norms, respectively, for $$p,q\in [1,\infty ]$$ satisfying $$1/p+1/q=1$$ . In addition, the deltoidal norm and its dual norm are shown to have better proximity to the $$\ell _p$$ -norm than the CVaR norm and its dual. Numerical examples demonstrate that LP solutions with optimized parameters attain better approximation of the $$\ell _{2}$$ -norm than the $$\ell _1$$ - and $$\ell _\infty $$ -norms do. [ABSTRACT FROM AUTHOR]