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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]

We prove the existence of functions that extremize the endpoint L 2 to L 4 adjoint Fourier restriction inequality on the one-sheeted hyperboloid in Euclidean space R 4 and that, taking symmetries into consideration, any extremizing sequence has a subsequence that converges to an extremizer. [ABSTRACT FROM AUTHOR]

In this paper we will establish a new Pólya–Szegö type inequality for a weighted gradient of a function on R 2 with respect to a weighted area. In order to do that we need to study an isoperimetric problem for the weighted area. We then apply the inequality to prove embedding theorems for weighted Sobolev spaces and to calculate the best constant in the Sobolev imbedding theorems. In our upcoming manuscript the obtained results in this note will be used to study boundary value problems for semilinear degenerate elliptic equations, see Luyen et al. (). [ABSTRACT FROM AUTHOR]

Ptolemy's inequality is a classic relationship between the distances among four points in Euclidean space. Another relationship between six distances is the 4-point condition, an inequality satisfied by the lengths of the six paths that join any four points of a metric (or weighted) tree. The 4-point condition also characterizes when a finite metric space can be embedded in such a tree. The curious observer might realize that these inequalities have similar forms: if one replaces addition and multiplication in Ptolemy's inequality with maximum and addition, respectively, one obtains the 4-point condition. We show that this similarity is more than a coincidence. We identify a family of Ptolemaic inequalities in CAT-spaces parametrized by a real number and show that a certain limit involving these inequalities, as the parameter goes to negative infinity, yields the 4-point condition, giving an elementary proof that the latter is the tropicalization of Ptolemy's inequality. [ABSTRACT FROM AUTHOR]

Ball's complex plank theorem states that if v 1 , ⋯ , v n are unit vectors in C d , and t 1 , ⋯ , t n are non-negative numbers satisfying ∑ k = 1 n t k 2 = 1 , then there exists a unit vector v in C d for which | ⟨ v k , v ⟩ | ≥ t k for every k. Here we present a streamlined version of Ball's original proof. [ABSTRACT FROM AUTHOR]

Inclusion properties are studied for balls of the triangular ratio metric, the hyperbolic metric, the j ∗ -metric, and the distance ratio metric defined in the unit ball domain. Several sharp results are proven and a conjecture about the relation between triangular ratio metric balls and hyperbolic balls is given. An algorithm is also built for drawing triangular ratio circles or three-dimensional spheres. [ABSTRACT FROM AUTHOR]

We present generalizations of Carnot's theorem and of the classical Erdös–Mordell inequality. [ABSTRACT FROM AUTHOR]

We consider the punctured plane with volume density | x | α and perimeter density | x | β . We show that centred balls are uniquely isoperimetric for indices (α , β) which satisfy the conditions α - β + 1 > 0 , α ≤ 2 β and α (β + 1) ≤ β 2 except in the case α = β = 0 which corresponds to the classical isoperimetric inequality. As an application, we verify a conjecture due to Caldiroli and Musina relating to the best constant in the Caffarelli–Kohn–Nirenberg inequality. [ABSTRACT FROM AUTHOR]

Recently, Dinew and Popovici introduced and studied an energy functional F acting on the metrics in the Aeppli cohomology class of a Hermitian-symplectic metric and showed that in dimension 3 its critical points (if any) are Kähler. In this article we further investigate the critical points of this functional in higher dimensions and under holomorphic deformations. We first prove that being a critical point for F is a closed property under holomorphic deformations. We then show that the existence of a Kähler metric ω in the Aeppli cohomology class is an open property under holomorphic deformations. Furthermore, we consider the case when the (2, 0)-torsion form ρω 2, 0 of ω is ∂-exact and prove that this property is closed under holomorphic deformations. Finally, we give an explicit formula for the differential of F when the (2, 0)-torsion form ρω2, 0 is ∂-exact.

In this article, the main objective is construction of fractional Newton–Simpson-type inequalities with the concept of majorization. We established a new identity on estimates of definite integrals utilizing majorization and this identity will lead us to develop new generalized forms of prior estimates. Some basic inequalities such as Hölder's, power-mean, and Young's along with the Niezgoda–Jensen–Mercer inequality have been used to obtain new bounds and it has been determined that the main findings are generalizations of many results that exist in the literature. Applications to the quadrature rule are given as well. We make links between our findings and a number of well-known discoveries in the literature. [ABSTRACT FROM AUTHOR]

Anisotropic generalized Orlicz spaces have been investigated in many recent papers, but the basic assumptions are not as well understood as in the isotropic case. We study the greatest convex minorant of anisotropic Φ -functions and prove the equivalence of two widely used conditions in the theory of generalized Orlicz spaces, usually called (A1) and (M). This provides a more natural and easily verifiable condition for use in the theory of anisotropic generalized Orlicz spaces for results such as Jensen's inequality which we obtain as a corollary. [ABSTRACT FROM AUTHOR]

We introduce an experimental version of GeoGebra that successfully conjectures and proves a large scale of geometric inequalities by providing an easy-to-use graphical interface. GeoGebra Discovery includes an embedded version of the Tarski/QEPCAD B system which can solve a real quantifier elimination problem, so the input geometric construction can be translated into a semi-algebraic system, and after some algebraic manipulations, the obtained formula can be translated back to a precisely stated geometric inequality. We provide some non-trivial examples to illustrate the performance of GeoGebra Discovery when dealing with inequalities, as well as some technical difficulties. [ABSTRACT FROM AUTHOR]

In the article, we prove 18 inequalities involving inradius, a length of one side and one additional element of a given triangle. 14 of these inequalities are the necessary and sufficient conditions for the existence and uniqueness of such a triangle. All proofs are based on standard methods of calculus and can serve as a good demonstration of the relationship between different branches of mathematics (geometry, algebra, trigonometry, calculus). The article can be used by teachers and students in courses on advanced classical geometry. [ABSTRACT FROM AUTHOR]

In Donnay's and Van Brummelen's monographs on spherical trigonometry, the Cesàro method is revitalized to derive various results on spherical triangles. Using Cesàro's triangles, we derive in this paper some further results on spherical triangles that are not given in these books. Among these are results concerning the relation of sides of a spherical triangle and their opposite angles, Lexell's theorem, and a result concerning the area of spherical triangles with one variable side-length. These results are applied to derive theorems of isoperimetric type for spherical polygons and the corresponding extremal properties of regular spherical polygons. [ABSTRACT FROM AUTHOR]

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 initiate the study of the higher-order Escobar constants I k (M) , k ≥ 3 , on bounded planar domains M. The Escobar constants I k of the unit disk and a family of polygons are provided. [ABSTRACT FROM AUTHOR]

We show that a smooth embedded paper Moebius band must have aspect ratio at least λ 1 = 2 4 - 2 3 + 4 3 4 2 + 2 2 3 - 3 = 1.69497 ... This bound comes more than 3/4 of the way from the old known bound of π / 2 = 1.5708 ... to the conjectured bound of 3 = 1.732 ... [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]

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]

In this paper, we establish two strengthened versions of Klamkin's inequality for an n-dimensional simplex in Euclidean space E n and give some applications. [ABSTRACT FROM AUTHOR]

A pair (α , β) of simple closed geodesics on a closed and oriented hyperbolic surface M g of genus g is called a filling pair if the complementary components of α ∪ β on M g are simply connected. The length of a filling pair is defined to be the sum of their individual lengths. In Aougab and Huang (Algebr Geom Topol 15:903–932, 2015), Aougab–Huang conjectured that the length of any filling pair on M g is at least m g 2 , where m g is the perimeter of the regular right-angled hyperbolic 8 g - 4 -gon. In this paper, we prove a generalized isoperimetric inequality for disconnected regions and we prove the Aougab–Huang conjecture as a corollary. [ABSTRACT FROM AUTHOR]

We study planar N-bubbles that minimize, under an area constraint, a weighted perimeter P ε {P_{\varepsilon}} depending on a small parameter ε > 0 {\varepsilon>0}. Specifically, we weight 2 - ε {2-\varepsilon} the boundary between the bubbles and 1 the boundary between a bubble and the exterior. We prove that as ε → 0 {\varepsilon\to 0} , minimizers of P ε {P_{\varepsilon}} converge to configurations of disjoint disks that maximize the number of tangencies, each weighted by the harmonic mean of the radii of the two tangent disks. We also obtain some information on the structure of minimizers for small ε. [ABSTRACT FROM AUTHOR]

Given a measurable set A ⊂ R d we consider the large-distance graph G A , on the ground set A, in which each pair of points from A whose distance is bigger than 2 forms an edge. We consider the problems of maximizing the 2d-dimensional Lebesgue measure of the edge set as well as the d-dimensional Lebesgue measure of the vertex set of a large-distance graph in the d-dimensional Euclidean space that contains no copies of a complete graph on k vertices. The former problem may be seen as a continuous analogue of Turán's classical graph theorem, and the latter as a "graph-theoretic" analogue of the classical isodiametric problem. Our main result yields an analogue of Mantel's theorem for large-distance graphs. Our approach employs an isodiametric inequality in an annulus, which might be of independent interest. [ABSTRACT FROM AUTHOR]

In this work we investigate the heat kernel of the Laplace–Beltrami operator on a rectangular torus and the according temperature distribution. We compute the minimum and the maximum of the temperature on rectangular tori of fixed area by means of Gauss' hypergeometric function 2 F 1 and the elliptic modulus. In order to be able to do this, we employ a beautiful result of Ramanujan, connecting hypergeometric functions, the elliptic modulus and theta functions. Also, we investigate the temperature distribution of the heat kernel on hexagonal tori and use Ramanujan's corresponding theory of signature 3 to derive analogous results to the rectangular case. Lastly, we show connections to the problem of finding the exact value of Landau's "Weltkonstante", a universal constant arising in the theory of extremal holomorphic mappings; and for a related, restricted extremal problem we show that the conjectured solution is the second lemniscate constant. [ABSTRACT FROM AUTHOR]

This paper stems from the idea of adopting a new approach to solve some classical optimal packing problems for balls. In fact, we attack this kind of problems (which are of discrete nature) by means of shape optimization techniques, applied to suitable Γ -converging sequences of energies associated to Cheeger type problems. More precisely, in a first step we prove that different optimal packing problems are limits of sequences of optimal clusters associated to the minimization of energies involving suitable (generalized) Cheeger constants. In a second step, we propose an efficient phase field approach based on a multiphase Γ -convergence result of Modica–Mortola type, in order to compute those generalized Cheeger constants, their optimal clusters and, as a consequence of the asymptotic result, optimal packings. Numerical experiments are carried over in two and three space dimensions. Our continuous shape optimization approach to solve discrete packing problems circumvents the NP-hard character of these ones, and efficiently leads to configurations close to the global minima. [ABSTRACT FROM AUTHOR]

We show the counter-intuitive fact that some weighted isoperimetric problems on the half-space R + N , for which half-balls centered at the origin are stable, have no solutions. A particular case is the measure d μ = x N α d x , with α ∈ (- 1 , 0) . Some results on stability and nonexistence for weighted isoperimetric problems on R N are also obtained. [ABSTRACT FROM AUTHOR]

We consider a class of isoperimetric problems on , where the volume and the area element carry two different weights of the type. We solve them in a special case while a more detailed study is contained in Alvino et al. (The isoperimetric problem for a class of non-radial weights and applications, in preparation). Our results imply a weighted Pólya–Szëgo principle and a priori estimates for weak solutions to a class of boundary value problems for degenerate elliptic equations. [ABSTRACT FROM AUTHOR]

We compute the Cheeger constant of tubular neighbourhoods of complete curves in an arbitrary dimensional Euclidean space and raise a question about curved spherical shells. [ABSTRACT FROM AUTHOR]

The paper is devoted to a new family of variational problems for differential inclusions, motivated by the protection of human and wildlife habitats when an invasive environmental disaster occurs. Indeed, the mathematical model consists of a differential inclusion describing the expansion of invaded region over time, and an artificial barrier that cannot be penetrated by the invasive agent is erected to shield the habitat, which serves as the control strategy and is characterized as a one-dimensional rectifiable set. We consider an isotropic case that the disaster spreads with uniform speed in all directions, and develop an equivalence result on the minimum construction speed required for implementing a sheltering strategy determined by a rectifiable Jordan curve. By measuring the invaded portion of barriers over time, it is shown that each connected non-convex habitat admits an equivalent habitat that contains the given habitat and their minimum speeds are equal. In particular, the boundary of an equivalent habitat can be partitioned into two arcs: an arc is locally convex, and the other is part of the boundary of illuminated area. This leads to a corollary on the existence of admissible sheltering strategies for non-convex habitats. [ABSTRACT FROM AUTHOR]

We prove that Demailly’s holomorphic Morse inequalities hold true for complex orbifolds by using a heat kernel method. Then we introduce the class of Moishezon orbifolds and as an application of our inequalties, we give a geometric criterion for a compact connected orbifold to be a Moishezon orbifolds, thus generalizing Siu’s and Demailly’s answers to the Grauert-Riemenschneider conjecture to the orbifold case. [ABSTRACT FROM AUTHOR]

The Kneser-Poulsen conjecture says that if a finite collection of balls in a Euclidean (spherical or hyperbolic) space is rearranged so that the distance between each pair of centers does not increase, then the volume of the union of these balls does not increase as well. We give new results about central sets of subsets of a Riemannian manifold and apply these results to prove new special cases of the Kneser-Poulsen conjecture in the two-dimensional sphere and the hyperbolic plane. [ABSTRACT FROM AUTHOR]

We establish a sharp affine $$L^p$$ Sobolev trace inequality by using the $$L_p$$ Busemann-Petty centroid inequality. For $$p = 2$$ , our affine version is stronger than the famous sharp $$L^2$$ Sobolev trace inequality proved independently by Escobar and Beckner. Our approach allows also to characterize all extremizers in this case. For this new inequality, no Euclidean geometric structure is needed. [ABSTRACT FROM AUTHOR]

We find a monotone quantity along the inverse mean curvature flow and use it to prove an Alexandrov-Fenchel-type inequality for strictly convex hypersurfaces in the n-dimensional sphere, $$n \ge 3$$ . [ABSTRACT FROM AUTHOR]

We study the linear polarization constants of finite dimensional Banach spaces. We obtain the correct asymptotic behaviour of these constants for the spaces [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]

We investigate a potential obtained as the convolution of a radially symmetric function and the characteristic function of a body (the closure of a bonded open set) with exterior cones. In order to restrict the location of a maximizer of the potential into a smaller closed region contained in the interior of the body, we give an estimate of the potential using the exterior cones of the body. Moreover, we apply the result to the Poisson integral for the upper half-space. [ABSTRACT FROM AUTHOR]