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## Due May 17

### Problem 1

Let $s\ge 0$ and $\mathcal H^s$ the Hausdorff measure with exponent $s$ in $\mathbb R^d$.
1. If $A\subset B$, then $\mathcal H^s(A) \le \mathcal H^s(B)$.
2. If $A = \bigcup_j A_j$, then $\displaystyle \mathcal H^s(A) \le \sum_j \mathcal H^s(A_j)$.
3. If $\text{dist}(A,B)>0$, then $\mathcal H^s(A\cup B) = \mathcal H^s(A) + \mathcal H^s(B)$.

### Problem 2

If $A\subset\mathbb R^d$ is countable, then $\dim(A) = 0$.

### Problem 3

If $0 < p < 1$, the function $x\mapsto x^p$ is concave: for all $x,y>0$ and $t\in[0,1]$,
$(tx + (1-t)y)^p \ge t x^p + (1-t) y^p$.

### Problem 4

Let $f:\mathbb R^d \to \mathbb R^d$ be a similitude with coefficient $\alpha >0$: for every $x,y\in\mathbb R^d$,
$|f(x) - f(y)| = \alpha |x-y|$.
Let $g(x) = \dfrac{1}{\alpha} (f(x) - f(0))$.
1. For all $x,y\in\mathbb R^d$, $g(x)\cdot g(y) = x\cdot y$.
2. If $e_1, e_2,\ldots,e_d$ is the standard basis for $\mathbb R^d$, then $g(e_1),g(e_2),\ldots,g(e_d)$ is an orthonormal basis for $\mathbb R^d$.
3. For $x,y\in\mathbb R^d$ and $j=1,2,\ldots,d$,
$g(x+y)\cdot g(e_j) = (g(x) + g(y))\cdot g(e_j)$.
4. For $x\in\mathbb R^d$, $\lambda\in\mathbb R$ and $j=1,2,\ldots,d$,
$g(\lambda x)\cdot g(e_j) = \lambda g(x)\cdot g(e_j)$.
5. $g$ is an orthogonal linear transformation.
Conclude that every similitude in $\mathbb R^d$ is of the form
$f(x) = \alpha Mx + x_0$,
where $\alpha>0$, $M\in O(d)$ and $x_0\in\mathbb R^d$.

### Problem 5

The polynomial $x^3 + 2x - 1$ has a unique real root, and the root is in $(0,\gamma)$, where
$\displaystyle \gamma = \frac{1}{\varphi} = \frac{\sqrt 5 - 1}{2}.$