Linear Threshold for Oertel's Conjecture on the Mixed-Integer Volume

Abstract

Grünbaum's inequality guarantees that the centroid of a convex body has halfspace depth at least $1/e$: every halfspace containing the centroid captures at least a $1/e$ fraction of the body's volume. For mixed-integer convex sets $S=C\cap(\mathbb Z^n\times\mathbb{R}^d)$ where $C$ is a convex body, Oertel conjectured that there exists $y\in S$ such that every closed halfspace $H$ containing $y$ satisfies $H_d(S\cap H)\ge \frac{1}{2^ne} H_d(S)$, where $H_d(X)$ denotes the $d$-dimensional Hausdorff measure of $X$. This conjecture is closely connected to complexity bounds for cutting plane methods and information complexity in mixed-integer convex optimization. Basu and Oertel established this conjecture for sets of sufficiently large lattice width, with the required lower bound on lattice width depending exponentially on the dimension. More recently, Cristi and Salas reduced this threshold to a polynomial one by assuming that $\mathrm{proj}_{\mathbb{R}^n}(C)$ contains a Euclidean ball of radius at least $1178 d^2n^{3/2}$. We prove that when the projection contains an $\ell_\infty$ ball of radius $k\ge(3e/2)(n+d)$, which is linear in the dimensions $n$ and $d$, there exists $y^*\in S$ such that every closed halfspace $H$ containing $y^*$ satisfies $H_d(S\cap H)\ge\left(\frac{1}{e}-\frac{3(n+d)}{2k}\right)H_d(S)$. In particular, when $k\ge 3e(n+d)$ we obtain $H_d(S\cap H)\ge \frac{1}{2e}H_d(S)\ge \frac{1}{2^ne}H_d(S)$, thus verifying Oertel's conjecture for a significantly larger family of sets than previous results. We also show that this linear scaling is necessary: when the radius is sublinear in the total dimension, the maximum achievable halfspace depth can be arbitrarily small relative to the total mixed-integer volume, so no dimension-independent constant fraction lower bound is possible under such an assumption alone. However, the conjecture remains open in its full generality.

Figures (6)

• Figure 1: Geometric view (shown over $\mathbb{Z}^2\times\mathbb{R}$) of the superlevel set $L_s=\left\{\mathbf{z}\in\mathbb{R}^2:\ g(\mathbf{z})\ge s\right\}$ and the sets $L_s+Q$ and $L_s\ominus Q$. At level $s$, the union $\bigcup_{\mathbf{z}\in L_s\cap\mathbb{Z}^2}(\mathbf{z}+Q)$ has $\mathcal{H}_2$ measure $\#(L_s\cap\mathbb{Z}^2)$ and satisfies $L_s\ominus Q\subseteq \bigcup_{\mathbf{z}\in L_s\cap\mathbb{Z}^2}(\mathbf{z}+Q)\subseteq L_s+Q$, which is the geometric content of Lemma \ref{['lem:lattice-sandwich']}.
• Figure 2: One dimensional restriction along an affine line in $\mathbb{R}^n$ of the envelope identities in Lemma \ref{['lem:q-envelopes']}. The blue dashed curve is $g$, and the gold and green curves are $g_+$ and $g_-$. The red columns correspond to unit cells centered at integer points on the slice. Equation \ref{['eq:level-identities']} identifies the superlevel sets of $g_+$ and $g_-$ with $L_s+Q$ and $L_s\ominus Q$, which are integrated in \ref{['eq:int-thick']} and \ref{['eq:int-erode']}.
• Figure 3: Geometric view (shown for $n=d=1$) of the symmetrized body $E=\left\{(\mathbf{z},\mathbf{x})\in \mathbb{R}^n\times\mathbb{R}^d:\ \mathbf{z}\in K,\ \|\mathbf{x}\|_2\le g(\mathbf{z})\right\}$ (blue) together with its Minkowski thickening $E^+=E+(Q\times\{\mathbf{0}\})$ (gold) and erosion $E^-=E\ominus(Q\times\{\mathbf{0}\})$ (green). For $\mathbf{z}\in \mathbb{R}^n$, the $\mathbf{x}$ fibers of $E$, $E^+$, and $E^-$ are centered Euclidean balls whenever they are nonempty, with radii given by $g(\mathbf{z})$, $g_+(\mathbf{z})$, and $g_-(\mathbf{z})$ from \ref{['eq:gpm']}. The dashed sets depict the dilations $(1\pm\epsilon)E$ with $\epsilon=1/(2k)$, and \ref{['eq:contain-thick']} and \ref{['eq:contain-erode']} compare these dilations with $E^+$ and $E^-$.
• Figure 4: Centroid rounding in Lemma \ref{['lem:centroid-rounding']} (shown for $n=d=1$). The centroid $\mathfrak{c}(C)$ has $\mathbf{z}$ projection $\mathbf{z}_C\notin\mathbb{Z}^n$, and shrinking toward a point $\mathbf{w}^\star\in C$ with $\epsilon=1/(2k)$ yields $C'=(1-\epsilon)C+\epsilon\mathbf{w}^\star$. By affine equivariance, $\mathfrak{c}(C')=(1-\epsilon)\mathfrak{c}(C)+\epsilon\mathbf{w}^\star$, and $\mathbf{w}^\star$ is chosen so that $\mathop{\mathrm{proj}}\nolimits_{\mathbb{R}^n}(\mathfrak{c}(C'))=\mathbf{z}^\star\in\mathbb{Z}^n$. The vertical lines indicate the fibers $\mathbb{Z}\times\mathbb{R}$.
• Figure 5: Schematic for Lemma \ref{['lem:half-bisect']}. A hyperplane $\partial H(c)$ orthogonal to $\mathbf{a}$ cuts $D$. The sets $D^+=\tfrac{1}{2}D+\tfrac{1}{2}\mathbf{y}^+$ and $D^-=\tfrac{1}{2}D+\tfrac{1}{2}\mathbf{y}^-$ are the two half scale translates that correspond to the alternatives in the lemma.

Theorems & Definitions (33)

• definition 1: Halfspace depth and centerpoints
• theorem 1: Grünbaum
• theorem 2
• corollary 1
• corollary 2
• lemma 1: Comparison inequality
• lemma 2: Lattice sandwich
• proof
• lemma 3: Layer cake
• proof