Weighted norm
The weighted norm for a vector X = (x1, x2, …, xn) and weights W = (w1, w2, …, wn), where each wi ≥ 0 and wi > 0 for at least one i, is defined by
where w = the sum of the wi.
Weighted distance
The weighted Minkowski distance between vectors X and Y is ||X–Y||p,W.
If y is a scalar, then ||X–y||p,W = ||X–Y||p,W where Y is the vector all of whose elements are y.
For any p, 1 ≤ p ≤ 2, we can define
For p = 2, it turns out that Lp,W(X) is equal to the weighted mean. Similarly, for p = 1, Lp,W(X) is equal to the weighted median (see Weighted Mean and Median).
See Lp Estimators for how to calculate the weighted Minkowski distance and weighted Lp,W(X) values in Excel.
References
Zornoza, J. (2020) Distance metric for machine learning. Aigents
https://aigents.co/data-science-blog/publication/distance-metrics-for-machine-learning
Wikipedia (2020) Minkowski distance
https://en.wikipedia.org/wiki/Minkowski_distance