Triangular inequality for 3d Euclidean simplicial complex in loop quantum gravity

Triangular inequality is an important relation in geometry such that this relation, intuitively, is a statement that the direct line connecting two points is the shortest one. Loop quantum gravity is presented after a reformulation of gravity using Ashtekar variables. The quantization follows the Dirac procedures, which results in the existence of state of quanta of 3d space as an element of Hilbert space. Spin network states has become the basis state for quanta of space in loop quantum gravity. In loop quantum gravity space is discrete and the geometrical quantity is quantized at the Planck scale. In 3d space, we can define triangular discretization of the hypersurface. In this article we discuss the length spectrum and check whether the triangular inequality is satisfied by the quantum length. The answer to the question is positive, such that even at the Planck scale the triangular inequality is still valid.