This is an addition to Noam's answer. In higher dimensions the same type of reasoning---using explicit conditions on the coefficients of reduced quadratic forms---is not feasible. Nevertheless, it is possible to prove the following result:

**Theorem.** Let $E \subset \mathbb{R^n}$ be an $n$-dimensional ellipsoid centered at the origin and containing no other integer point. There exists a transformation $T \in GL(n,\mathbb{Z})$ such that $T(E)$ is contained in the ball of radius
$$
\left(\frac{3}{2}\right)^{(n-1)(n-2)/2} \frac{2^n}{\epsilon_n}\sqrt{n^{n-1}}
$$
centered at the origin.

Here $\epsilon_n$ is the volume of the unit ball of dimension $n$.

This theorem follows easily from the following two results:

**Theorem.** Given a positive-definite $n \times n$ matrix $A = (a_{ij})$, there exists a unimodular integer matrix $U$ such that the matrix $A' = (a'_{ij})$
defined by $A' := U^t A U$ satisfies the following inequality:
$$
\left(\frac{2}{3}\right)^{(n-1)(n-2)} \left(\frac{\epsilon_n}{2^n}\right)^2 \leq \frac{\det(A')}{a'_{11} a'_{22} \cdots a'_{nn}} .
$$

**Theorem.** If $A = (a_{ij})$ is a positive-definite $n \times n$ matrix, then for every $x \in \mathbb{R}^n$ we have that
$$
\frac{\det(A)}{n^{n-1}a_{11} a_{22} \cdots a_{nn}} \sum a_{ii} x_i^2 \leq \sum a_{ij} x_i x_j \leq n \sum a_{ii} x_i^2 .
$$

The first of these two results is an important theorem of Minkowski on the reduction of positive quadratic forms (see Theorem 3 in page 69 of Lekkerkerker), while the second is an improved version of Lemma 4 in page 43 of Siegel's Lectures on
Quadratic Forms.