# Tagged Questions

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### Looking for a proof that the diameter of the smallest bounding circle is less than or equal to $\frac{2}{\sqrt{3}}$ times the diameter of the set

This came up while I was attempting to solve an old journal problem. It's not the easiest result to search for so I figured I would ask. Let $E$ be a subset of $\mathbb{R}^2$, then the diameter of ...
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### In regards to metric spaces, does $d^\star$ have an accepted name, or notation? Do any authors use it?

(I write $\omega$ for the set $\{0,1,2,\ldots\}$.) Let $X$ denote a metric space with metric $d$. Define a function $d^{\star} : X^\omega \times X^\omega \rightarrow [0,\infty]^\omega$ by writing ...
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### Gromov hyperbolic metric spaces are quasi-convex

I'm aware about the fact stated above, but I'm not able to find some references or proofs besides Gromov's Hyperbolic Groups - Essays in Group Theory. I'll state things precisely. I will consider a ...
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### $d(x,y)^2 \le d(x,z)^2 - d(y,z)^2\color{}{+\varphi\big(x,y,z,d(x,y),d(y,z) \big)}?$

In a metric space $(M,d)$ the triangle inequality $d (x, z) \le d(x, y) + d (y, z)$ gives us's the inequalitie $$\quad d(x,y)^2 \ge d(x,z)^2 - d(y,z)^2\;\color{}{{-2\cdot d(x,y)\cdot d(y,z)}}$$ ...
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### What is the metric spaces needed to motivate concepts of general topology?

I intend to start learning some topology on my own. I wonder How much metric spaces I should know in order to motivate the concepts of topology? I know it's possible to learn topology without any ...
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### Is there a name for this partial order between metrics?

Suppose we have a set $X$ and two metrics $d_1,d_2$ on it (which may or may not attain $\infty$). Assume furthermore that $d_1,d_2$ have the same metric components (where a metric comoponent is a ...
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### Proving that a function is a metric

Let $$p(x,y)= \left|\frac{1}{x} - \frac{1}{y}\right|$$ for $x,y > 0$. Prove that $p$ is a metric for $(0,\infty)$. This question is from Methods of Real Analysis, 2nd edition by Richard ...
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### book for metric spaces

Can anybody suggest me a good book on Metric Spaces. Although I am not new to this subject, but want to polish my knowledge. I want a book which can clearly clear my basics. I want to start from the ...