Asymptotic behavior of the Manhattan distance in $n$-dimensions: Estimating multidimensional scenarios in empirical experiments

Understanding distance metrics in high-dimensional spaces is crucial for various fields such as data analysis, machine learning, and optimization. The Manhattan distance, a fundamental metric in multi-dimensional settings, measures the distance between two points by summing the absolute differences along each dimension. This study investigates the behavior of Manhattan distance as the dimensionality of the space increases, addressing the question: how does the Manhattan distance between two points change as the number of dimensions n increases?. We analyze the theoretical properties and statistical behavior of Manhattan distance through mathematical derivations and computational simulations using Python. By examining random points uniformly distributed in fixed intervals across dimensions, we explore the asymptotic behavior of Manhattan distance and validate theoretical expectations empirically. Our findings reveal that the mean and variance of Manhattan distance exhibit predictable trends as dimensionality increases, aligning closely with theoretical predictions. Visualizations of Manhattan distance distributions across varying dimensionalities offer intuitive insights into its behavior. This study contributes to the understanding of distance metrics in high-dimensional spaces, providing insights for applications requiring efficient navigation and analysis in multi-dimensional domains.