Homework 3 (due 4-Feb)

The following problems are due in class on 4-Feb.

Part I: Convexity

For these problems, refer to the example proof done in class on the convexity of the unit ball.

1. Let $U=[0,1]\times[0,1]$ be the unit square in the Euclidean plane $\mathbb{R}^2$. Show that $U$ is convex, by demonstrating that for any two points $x\in U$ and $y\in U$, $(1-t)x+ty\in U$ for all $t$ between 0 and 1. Then show that in $\mathbb{R}^n$ the set $[0,1]\times[0,1]\times\cdots\times[0,1]$ is also convex.

2. Complete problem 26 in chapter 1.

Part II: Inequalities.

3. Show that the triangle inequality generalizes: for any k vectors $\vec x_1, \cdots, \vec x_k\in\mathbb{R}^n$,

(1)
\begin{align} \big|\vec x_1+\vec x_2+\cdots+\vec x_k\big|\leq |\vec x_1| + |\vec x_2| + \cdots + |\vec x_k|. \end{align}

4. Find an example of vectors $\vec x$ and $\vec y$ where the Cauchy-Schwarz Inequality is an equality, i.e., $\langle\vec x,\vec y\rangle=|\vec x|\cdot|\vec y|$. Make a hypothesis regarding what conditions are required in general for this equality.

5. For any $\vec x,\vec y\in\mathbb{R}^n$, show that $\big|\vec x+\vec y\big|^2+\big|\vec x-\vec y\big|^2=2|\vec x|^2+2|\vec y|^2$. If $\vec x$ and $\vec y$ are two sides of a parallelogram in $\mathbb{R}^2$, what does this fact imply about the parallelogram?

6. If $n\geq 2$ and $\vec x\in\mathbb{R}^n$, find a nonzero vector $\vec y\in\mathbb{R}^n$ such that $\vec x\cdot\vec y=0$. Show that this is impossible if $n=1$.

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