The way k-means is constructed is not based on distances.

K-means minimizes within-cluster variance. If you look at the definition of variance, it is identical to the sum of squared Euclidean distances from the centroid

Do you lose convergence guarantees if you minimize "sum of squared distances from the centroid" using your other-than-Euclidean distance metric?

Suppose you use prices and square meters, to cluster apartments. Suppose you also use a simple metric, like:

  distance <- sqrt(d_price*2 + d_area*2)

Without normalizing the units somehow, so that they have a meaningfully similar numeric expression, you are essentially only clustering by price, which is in the tens- or hundreds-of-thousands.

You use very little information about the area, because that is in the hundreds, at most thousands.

This has nothing to do with using non-Euclidean distance, though. This is a separate issue related to having features on different scales.

It was an example metric with coefficients of 1 for each feature.

The point I was making was that the different scale of the features influences the result by a lot.

That is not a proper distance function.

What do you mean? Is it not the Euclidean distance?

https://en.wikipedia.org/wiki/Euclidean_distance