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I have a cluster $C$ of points in a high dimensional space $\mathbb R^n$. I want to know whether this cluster is convex or not. How can i study this? I thought of projecting $C$ on $\mathbb R^2$ to visualize it using Mapper or PCA.

If the projection of $C$ on $\mathbb R^2$ is convex, can i assume that it is convex? On the other hand, if the projection of $C$ is non convex, can i assume that $C$ is non convex?

Thank you!

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    $\begingroup$ Because convexity is an affine geometric property, affine transformations (which is what PCA uses) preserve it. $\endgroup$
    – whuber ♦
    Commented Jul 3, 2019 at 14:39
  • $\begingroup$ Thank you! one more stupid question: does the same works with non-convexity? In other words, is the image of a non-convex space non-convex? Thanks again! $\endgroup$
    – Alfred
    Commented Jul 3, 2019 at 15:36
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    $\begingroup$ Same answer for the same reason: any test of non-convexity will yield the same result when any (nonsingular) affine transformation is applied to the points. $\endgroup$
    – whuber ♦
    Commented Jul 3, 2019 at 15:36

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It's a linear projection.

You can trivially prove that it preserves convexity by this: if x is a point on the line between y and z, then there is a scalar a such that x=ax+(1-a)y. But since our map f is linear, then f(x)=f(ax+(1-a)y)=a f(x) + (1-a) f(y).

More precisely it's affine, but the key point is that there is non-linear part.

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