Geometric constant is one of the important tools to study geometric properties of Banach spaces. In this paper, we will introduce two new geometric constants JL(X) and YJ(X) in Banach spaces, which are symmetric and related to the side lengths of inscribed equilateral triangles of unit balls. The upper and lower bounds of JL(X) and YJ(X) as well as the values of JL(X) and YJ(X) for Hilbert spaces and some common Banach spaces will be calculated. In addition, some inequalities for JL(X), YJ(X) and some significant geometric constants will be presented. Furthermore, the sufficient conditions for uniformly non-square and normal structure, and the necessary conditions for uniformly non-square and uniformly convex will be established.

We investigate the Hyers–Ulam stability of an equation involving a single variable of the form ∥f(x)−αf(kn(x))−βf(kn+1(x))∥⩽u(x) where f is an unknown operator from a nonempty set X into a Banach space Y, and it preserves the addition operation, besides other certain conditions. The theory is employed and stability theorems are proven for various functional equations involving several variables. By comparing this method with the available techniques, it was noticed that this method does not require any restriction on the parity, on the domain, and on the range of the function. Our findings suggest that it is very much easy and more appropriate to apply the proposed method while investigating the stability of functional equations, in particular for several variables.

In this paper, we firstly investigate the constant H(X) proposed by Gao further by discussing several properties of it that have not yet been discovered. Secondly, we focus on a new constant GL(X) closely related to H(X), along with a variety of geometric properties. In addition, we show several relations among it and the several basic geometric constants via a few inequalities. Finally, we manage to characterize the geometric properties of its generalized forms GL(X,p) and CL(X) explicitly.

In this paper, we will introduce a new geometric constant LYJ(λ,μ,X) based on an equivalent characterization of inner product space, which was proposed by Moslehian and Rassias. We first discuss some equivalent forms of the proposed constant. Next, a characterization of uniformly non-square is given. Moreover, some sufficient conditions which imply weak normal structure are presented. Finally, we obtain some relationship between the other well-known geometric constants and LYJ(λ,μ,X). Also, this new coefficient is computed for X being concrete space.

We introduce a new geometric constant Jin(X) based on a generalization of the parallelogram law, which is symmetric and related to the length of the inscribed quadrilateral side of the unit ball. We first investigate some basic properties of this new coefficient. Next, it is shown that, for a Banach space, Jin(X) becomes 16 if and only if the norm is induced by an inner product. Moreover, its properties and some relations between other well-known geometric constants are studied. Finally, a sufficient condition which implies normal structure is presented.