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 $\vec x_1, \cdots, \vec x_k\in\mathbb{R}^n$,

**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 $n=1$.