Successive minima (awaiting review)

Successive minima $\lambda_1,\dots,\lambda_n$ of a $\boldsymbol 0$-symmetric convex body $K \subset \mathbb R^n$ with respect to a full-rank lattice $L \subset \mathbb R^n$ is defined as $\lambda_i = \min \{ t > 0 : \text{dim}_{\mathbb R} \text{span}_{\mathbb R}(tK \cap L) \geq i \}$. Then $0 < \lambda_1 \leq \dots \leq \lambda_n$, and $$\frac{2^n \text{det}(L)}{n! \text{Vol}(K)} \leq \lambda_1 \cdots \lambda_n \leq \frac{2^n \text{det}(L)}{\text{Vol}(K)},$$ by Minkowski's Successive Minima Theorem. Successive minima of $L$ are the successive minima of the unit ball with respect to $L$.