A generalized complexity formalism for $$(n+2)$$-dimensional spacetime

In this work, we extend Herrera’s concept of complexity for spherically symmetric self-gravitating static fluid distributions to arbitrary $$(n+2)$$ ( n + 2 ) -dimensional spacetimes. By performing the orthogonal splitting of the Riemann tensor in higher dimensions, we derive a generalized scalar measure that quantifies the combined effects of energy density inhomogeneity and pressure anisotropy. This generalized complexity factor incorporates the dimensional dependence through the gravitational coupling constant and the geometry of the n n -sphere. We further analyze the vanishing complexity condition, identifying the specific balance between anisotropy and density gradients that yields structurally simple configurations in higher dimensions. Our results offer a foundational framework for investigating gravitational complexity in theories extending beyond four-dimensional general relativity.