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Q27E

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Found in: Page 437

### Linear Algebra and its Applications

Book edition 5th
Author(s) David C. Lay, Steven R. Lay and Judi J. McDonald
Pages 483 pages
ISBN 978-03219822384

# Question: 27. Give an example of a closed subset $$S$$ of $${\mathbb{R}^{\bf{2}}}$$ such that $${\rm{conv}}\,S$$ is not closed.

The set is $$S = \left\{ {\left( {x,y} \right):{x^2}{y^2} = 1,\,\,y > 0} \right\}$$.

See the step by step solution

## Step 1: Assume subset  $$S$$ such that $${\rm{conv }}S$$ is not closed

One of the possible sets is $$S = \left\{ {\left( {x,y} \right):{x^2}{y^2} = 1,\,\,y > 0} \right\}$$. This set is not closed as the equation $${x^2}{y^2} = 1,\,\,\,y > 0$$ is of a hyperbola in an upper half-plane that is shown below:

## Step 2: Check whether the assumed set $$S$$is in $${\mathbb{R}^{\bf{2}}}$$

The hyperbola $$xy = 1$$ opens upwards; that is, it satisfies all values of $$x$$ and for $$y > 0$$.

So, the set $$S$$ is in $${\mathbb{R}^{\bf{2}}}$$.