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Let X be a metric space and A a subset of X.

I would like to visualize this basic inequality: $$ \lvert d(x,A)-d(y,A) \rvert \le d(x,y) $$

I tried drawing A as a blob, but I couldn't draw something convincing. So I came up with A in the shape of a U: Equality case this is the case for equality, where everything is aligned. And the case for inequality: Inequality case

I guess this simplified scenario helps me visualize it, but I was wondering if a more general and convincing picture could be drawn with an arbitrary looking blob. The algebraic proof is simple enough to understand, I just feel like such an innocent looking inequality should have a simple picture, like the basic triangle inequality, but the fact that the points of A nearest to x and y can be different, makes it hard for me to draw this picture.

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  • $\begingroup$ What if $A = \{z\}$? $\endgroup$ Commented Nov 10 at 11:14

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Draw contours so that each contour represents the set of points whose distance to $A$ is constant. Then $|d(x,A) - d(y,A)|$ measures how far apart the contours passing through $x$ and $y$ are. In other words, it is proportional to the number of contour lines separating them.

The quantity $d(x,y)$ is the distance that point $x$ must move to reach $y$. But in order to reach $y$, the point $x$ must at least move from its own contour to the contour that passes through $y$. Since $|d(x,A) - d(y,A)|$ is the shortest possible displacement between those two contours, $d(x,y)$ cannot be smaller than $|d(x,A) - d(y,A)|$.

enter image description here

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  • $\begingroup$ Thank you for a very clear and concise illustration ! $\endgroup$ Commented Nov 10 at 12:07

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