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Show by an example that the correlation-based distance

$d(X,X^\prime)=1-\rho(X,X^\prime)=1-\frac{\sum_{j=1}^p (X_j-\bar{X})((X_j^\prime-\bar{X}^\prime)}{\sqrt(\sum_{j=1}^p (X_j-\bar{X})^2\sum_{j=1}^p (X_j^\prime-\bar{X}^\prime)^2)}$ where $X, X^\prime ∈ R^p$,

is not strictly speaking a metric.

My understanding: We need to show that distance $d$ violates the triangle inequality, but how can I show this using example. I am kinda confused! Thank you so much for your suggestions!

Quantam
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  • Since the triangle inequality concerns three points (random variables), it might help to understand the restrictions on correlation coefficients between multiple variables. Some relevant threads are https://stats.stackexchange.com/questions/72790, https://stats.stackexchange.com/questions/305441, https://stats.stackexchange.com/questions/445370, and https://stats.stackexchange.com/questions/256116. – whuber Apr 25 '21 at 14:16
  • @whuber What would be the appropriate example to show the violation of triangle inequality? I am kinda confused with the problem statement! – Quantam Apr 25 '21 at 14:19
  • Consider what the triangle inequality says and think about what combinations of correlation coefficients might make it false. It can help to consider extreme circumstances, such as when correlations are as positive or as negative as they can possibly be. It can also help to visualize correlations -- there are many ways to do so and likely you have encountered at least one. – whuber Apr 25 '21 at 14:22
  • There is a counter example for triangle inequality in [this answer](https://stats.stackexchange.com/a/135179/331313). – Adnan Baysal Aug 06 '21 at 19:32

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