TITLE: What does invariant to affine transformations mean?
QUESTION [2 upvotes]: I am researching multivariate medians. In one of my sources it is stated that "Liu showed that the simplicial median is invariant to affine transformations". I am a bit confused about what this means. I know that affine transformations preserve ratios of distances, collinearity but not necessarily angles or lengths. However, I don't know what it means for something to be invariant to affine transformations. Does it mean that it will always remain the same? Also I was wondering if there is a simple example to illustrate this to help improve my understanding.

REPLY [0 votes]: Affine transformation are a combination of a translation with a linear transformation
$$A(v)=Av+v_0$$
which preserves points, straight lines and planes.
Ratio by segments are preserved and in particular mean points are preserved by linearity
indeed if $$M=\frac{P+Q}{2}\implies A(M)=\frac{A(P)+A(Q)}{2}+v_0$$
Affine transformations
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