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Comment by klyrs | Hacker News Reader

Joris's paper discusses high-dimensional balls with various metrics (where the max-norm gives equality between the "ball" and "interval" concepts), but I skimmed your paper and it looks like you're doing only doing 1d. It isn't immediately obvious to me how one would perform computations on, say, 3-dimensional euclidean balls using your library. Is this something you've thought through (or was my skim too shallow)?

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maxnorm in 3-dimensions is not euclidean. It is however, a metric space (which is why they call them "balls"). Note that the following formula for vectors x and y over R is a valid metric (obeys triangle inequality) for _all_ n in _all_ dimensions, with n=2 being the usual euclidean metric:

    d(x, y) = |sum_i((x_i - y_i)^n)|^(1/n)

The maxnorm is the limit as n -> infinity, and is also a proper metric space. (the collection of these spaces are called Lp spaces)

Of course in one dimension, all of these Lp metrics are identical.

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

The library doesn't support this directly. What you can do for normed vector spaces over R (or C) is to use the number types of Arb for the coordinate vectors, implementing your own norm and metric functions for the vector space on top of this. It's fairly easy to do since Arb handles all rounding errors in the underlying real arithmetic.

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