Successive minima of a convex body

Let $\mathrm{\Lambda }\subseteq {\mathbb{R}}^n$ be a full lattice (Full Euclidean lattice and covolume) and let $C\subseteq {\mathbb{R}}^n$ be centrally symmetric (that is, $-C=C$), convex in the sense of A convex subset of ${\mathbb{R}}^m$ contains every line segment between two of its points, compact, and with nonempty interior. For a real $t>0$ write

$tC:=\{tx:x\in C\}\mathrm{.}$

For $1\le i\le n$ the $i$-th successive minimum of $C$ with respect to $\mathrm{\Lambda }$ is

${\lambda }_i(C,\mathrm{\Lambda }):=\inf \{t>0:{\dim }_{\mathbb{R}}{\mathrm{span}}_{\mathbb{R}}(tC\cap \mathrm{\Lambda })\ge i\},$

the infimum ranging over the nonempty set of real numbers $t>0$ for which the linear span of the finite set $tC\cap \mathrm{\Lambda }$ has dimension at least $i$. When no such $t$ exists the infimum is $+\mathrm{\infty }$; for the bodies considered here it is finite, as recorded in the remarks below. We write ${\lambda }_i$ for ${\lambda }_i(C,\mathrm{\Lambda })$ when $C$ and $\mathrm{\Lambda }$ are fixed in context, and we write ${\lambda }_0:=0$ by convention.

The scaling convention is that the body is enlarged and the lattice is held fixed: $tC\cap \mathrm{\Lambda }$ is the set of lattice points lying in the $t$-dilate of $C$. Equivalently, because $C=-C$, one may think of the shortest vectors of $\mathrm{\Lambda }$ in the norm whose unit ball is $C$.

The sets involved are finite. Each $tC$ is bounded because $C$ is compact, and a bounded subset of ${\mathbb{R}}^n$ meets a full lattice in finitely many points. Indeed, write $\mathrm{\Lambda }=B{\mathbb{Z}}^n$ using a basis matrix $B$. If $S\subseteq {\mathbb{R}}^n$ is bounded, choose $M>0$ with $\mathrm{\mid }x\mathrm{\mid }\le M$ for all $x\in S$. The inverse linear map is bounded by Every Euclidean linear map has a unique matrix and satisfies $\mathrm{\parallel }Lh{\mathrm{\parallel }}_2\le K\mathrm{\parallel }h{\mathrm{\parallel }}_2$ for some $K\ge 0$, say $\mathrm{\mid }{B}^{-1}x\mathrm{\mid }\le K\mathrm{\mid }x\mathrm{\mid }$. Thus if $Bm\in S$ for $m\in {\mathbb{Z}}^n$, then $\mathrm{\mid }m\mathrm{\mid }\le KM$, so every integer coordinate of $m$ lies in the finite interval $[-KM,KM]$; only finitely many such integer vectors occur. Consequently $\mathrm{span}(tC\cap \mathrm{\Lambda })$ is a finite-dimensional real subspace and the dimension in the definition is a genuine nonnegative integer, never an undecided quantity.

Monotonicity. If $0<s<t$ then $sC\subseteq tC$: for $x\in C$ one has $0\in C$ and $(s\mathrm{/}t)x+(1-s\mathrm{/}t)\cdot 0=(s\mathrm{/}t)x\in C$ by convexity, so $sx\in tC$. Hence $sC\cap \mathrm{\Lambda }\subseteq tC\cap \mathrm{\Lambda }$ and the dimension function is nondecreasing in $t$; the sets inside the infimum are therefore upward-closed, and $0<{\lambda }_1\le {\lambda }_2\le \cdots \le {\lambda }_n$.

Finiteness and positivity. Central symmetry and nonempty interior put the origin in the interior: if $v$ is an interior point then so is $-v$, and $0=(v+(-v))\mathrm{/}2$ is an interior point of $C$ by convexity. So $C$ contains a Euclidean ball $\rho B^n$ about the origin with $\rho >0$, and compactness of $C$ bounds it by some $R={\sup }_{x\in C}\mathrm{\mid }x\mathrm{\mid }<\mathrm{\infty }$, with $R>0$. Write $\mathrm{\Lambda }=\mathbb{Z}{b}_1\oplus \cdots \oplus \mathbb{Z}{b}_n$ and let $B:{\mathbb{R}}^n\to {\mathbb{R}}^n$ be the linear isomorphism $B(m)={\sum }_i{m}_i{b}_i$. Its inverse is a Euclidean linear map, so the bounded- linear-map result Every Euclidean linear map has a unique matrix and satisfies $\mathrm{\parallel }Lh{\mathrm{\parallel }}_2\le K\mathrm{\parallel }h{\mathrm{\parallel }}_2$ for some $K\ge 0$ gives a constant $K\ge 1$ such that $\mathrm{\mid }{B}^{-1}v\mathrm{\mid }\le K\mathrm{\mid }v\mathrm{\mid }$ for every $v\in {\mathbb{R}}^n$. For every nonzero integer vector $m\in {\mathbb{Z}}^n$, $\mathrm{\mid }m\mathrm{\mid }\ge 1$, and therefore $\mathrm{\mid }Bm\mathrm{\mid }\ge 1\mathrm{/}K$. If $Bm\in tC$, then $\mathrm{\mid }Bm\mathrm{\mid }\le tR$, so $t\ge 1\mathrm{/}(KR)$ and hence ${\lambda }_1\ge 1\mathrm{/}(KR)>0$. Conversely ${\lambda }_n<\mathrm{\infty }$: for each $i$ choose $t_i\ge \mathrm{\mid }{b}_i\mathrm{\mid }\mathrm{/}\rho$, so that $b_i\mathrm{/}{t}_i$ has norm at most $\rho$ and therefore lies in $C$; for $t={\max }_i{t}_i$ the set $tC\cap \mathrm{\Lambda }$ contains $b_1,\dots ,b_n$ and spans ${\mathbb{R}}^n$. The infima defining the ${\lambda }_i$ are thus positive and finite, and the next items prove that they are attained and control the lattice vectors at the attained levels.