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For points $\mathbf{a}, \mathbf{b}, \mathbf{x}, \mathbf{y} \in \mathbb{R}^d$ for some natural $d \ge 1$. I want to prove that if (Euclidean distance) $||\mathbf{x} - \mathbf{a}|| \le ||\mathbf{y} - \mathbf{a}||$ and $||\mathbf{x} - \mathbf{b}|| \le ||\mathbf{y} - \mathbf{b}||$, then for any point on line between $\mathbf{a}$ and $\mathbf{b}$, call it $\mathbf{c}$, $||\mathbf{x} - \mathbf{c}|| \le ||\mathbf{y} - \mathbf{c}||$

I know how to prove this equation using an argument derived from calculating the hyper-plane of equidistance points between $\mathbf{x}$ and $\mathbf{y}$, and then, using some more implications to argue that $\mathbf{a}$ and $\mathbf{b}$ must on the same side as the side of the hyperplane $\mathbf{x}$ because they are closer and then arguing that then $\mathbf{c}$ must be on the same side as the side of $\mathbf{x}$; however, I'm looking for a solution that directly derives the property that $||\mathbf{x}-\mathbf{c}|| \le ||\mathbf{y}-\mathbf{x}||$ using the properties of Euclidean distance.

I understand that we can parameterize $\mathbf{c}$ in terms of $\mathbf{a}$ and $\mathbf{b}$ as $\mathbf{c} = \alpha \mathbf{a} + (1 - \alpha)\mathbf{b}$ where $0 \le \alpha \le 1$, and that was my first step. Then, I tried to start from $||\mathbf{x}-\mathbf{c}||$ and reach that it is less than or equal to $||\mathbf{y} - \mathbf{c}||$ using a series of reformulations from my starting point and my parameterized definition of $\mathbf{c}$. However, I was not able to get anywhere. I'm quite stuck as to how to prove this in this manner, and any hints/advice would be greatly appreciated.

Thank you.

Somos
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dinodeep
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  • Have you already proved the triangle inequality for metric spaces? – nomen May 25 '22 at 17:46
  • Define $,d(\mathbf{a}):=||\mathbf{y}-\mathbf{a}||^2 - ||\mathbf{x}-\mathbf{a}||^2.,$ What is the relation between $,d(\mathbf{a}),$, $,d(\mathbf{b}),$ and $,d(\mathbf{c}),$? – Somos May 25 '22 at 17:48
  • The crucial fact is the convexity of the distance function on the line segment. Show that $|\mathbf c-\mathbf z|\le\min(\mathbf a-\mathbf z|,|\mathbf b-\mathbf z|)$. This is immediate from your parametric interpretation and the triangle inequality. – Ted Shifrin May 25 '22 at 17:50

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