How Does the Triangle Inequality Transform from Equality to Inequality?

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Discussion Overview

The discussion revolves around the transformation of the triangle inequality from equality to inequality, exploring the conditions under which this transformation occurs. Participants examine the implications of taking absolute values and squaring terms in the context of metric spaces.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • One participant questions how the equality in the triangle inequality transforms into an inequality, suggesting that taking the absolute value of 2uv may play a role due to its potential negative value.
  • Another participant agrees with the first claim, indicating that the first inequality arises from the properties of a metric space and references the Cauchy-Schwartz inequality.
  • A different participant inquires about the permissibility of squaring both sides of an inequality when both sides are non-negative, questioning why this would not hold if both sides were negative.
  • In response, a participant clarifies that the ability to square both sides relies on the positivity of the terms involved, providing an example to illustrate this point.

Areas of Agreement / Disagreement

Participants express varying levels of understanding regarding the transformation of the triangle inequality and the conditions for squaring terms. There is no consensus on the implications of squaring negative values, indicating an unresolved aspect of the discussion.

Contextual Notes

Limitations include assumptions about the positivity of terms and the conditions under which inequalities can be manipulated, which remain unresolved in the discussion.

Bashyboy
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Hello all,

I am currently reading about the triangle inequality, from this article
http://people.sju.edu/~pklingsb/cs.triang.pdf

I am curious, how does the equality transform into an inequality? Does it take on this change because one takes the absolute value of 2uv? Because before the absolute value, 2uv could be a negative value, thus making all of |u|^2 + 2uv + |v|^2 smaller, is this correct?
 
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You are correct ... that is why the first inequality appears. The second one is from the Cauchy-Schwartz inequality, as noted.

These are properties that are required for a metric space.
 
I have one other question. In the article, it says that since both sides of the inequality of non-negative, it is permissible to then square both sides of the inequality. Why would it not be possible to square both sides if both sides were negative?
 
I'm sure that they said "you can square each term since they are all positive". Try that with this inequality:

1 - 2 < 1 ... hence the requirement for all positive.
 

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