Diameter-Width Ratio in Planar Pseudo-Complete Sets

• Diameter-width ratio for planar pseudo-complete sets quantifies the link between a convex body’s maximum extent and minimal breadth within a Minkowski framework.
• It employs Minkowski asymmetry and symmetrization inequalities to derive sharp bounds, with extreme cases like the golden house achieving the maximal configuration.
• The analysis refines previous diameter-width limits by delineating distinct asymmetry regimes and explicit parameter domains that impact convex optimization.

A planar convex body’s diameter-width ratio provides a precise quantitative link between its maximum extent and its minimal breadth, measured in a Minkowski geometric framework. For pseudo-complete sets—convex bodies in $\mathbb{R}^2$ whose diameter cannot be increased without strictly enlarging the set—recent results yield an exact, asymmetry-sensitive bound. This bound is achieved by explicit descriptions involving the Minkowski asymmetry parameter and sharp symmetrization inequalities, characterizing the range of possible diameter-width ratios and specifying extremal and typical geometric configurations.

1. Foundational Concepts in Planar Minkowski Geometry

Let $C$ be a 0-symmetric convex body in $\mathbb{R}^2$, defining a gauge norm $\|\cdot\|_C$. For a convex compact set $K\subset\mathbb{R}^2$, the Minkowski diameter and width, both with respect to $C$, are

$$w(K,C) = 2\, r(K-K,\;C-C), \qquad D(K,C) = 2\, R(K-K,\,C-C),$$

where $r(\cdot,\cdot)$ and $R(\cdot,\cdot)$ denote inradius and circumradius relative to the given gauge. $K$ is pseudo-complete if $r(K,C) + R(K,C) = D(K,C)$. In planar geometry, this coincides with completeness and, when $C$ is "perfect" (notably Euclidean or in dimension 2), also with constant width.

The {\bf Minkowski asymmetry} of $K$, $s(K)$, is given by

$$s(K) := \inf \left\{ \lambda \ge 1\ :\ \exists c \in \mathbb{R}^2\ \text{such that } K-c \subset \lambda(c-K) \right\},$$

with equality $s=1$ if and only if $K$ is centrally symmetric and $s=2$ if and only if $K$ is a (nondegenerate) triangle.

2. Symmetrization and Critical Containment Parameters

Beyond $s(K)$, diameter-width analysis incorporates the containment parameter $\tau(K)$ for a Minkowski-centered convex compact set $K$. Translating $K$ so its Minkowski center is at the origin, two symmetrizations are considered:

• $K \cap (-K)$ (the "minimum")
• $(K-K)/2$ (the "arithmetic mean")

$\tau(K)$ is the minimal scaling such that

$K \cap (-K) \subset \tau(K)\, \frac{K-K}{2}.$

It is equivalently the reciprocal of the minimal radial ratio positioning the boundary of $K \cap (-K)$ inside $(K-K)/2$ via origin-centered homotheties: $$\frac{1}{\tau(K)} = \min_{v\in \partial(K\cap(-K))}\ \{\rho>0 : \rho v \in \partial ((K-K)/2) \}.$$ Always $1 \leq s(K) \leq 2$ and $2/(s(K)+1) \le \tau(K) \le c(s(K))$, with $c(s)$ an explicitly determined function.

3. Characterization of the Feasible Region: The $(s, \tau)$ Domain

A complete description of all possible parameter pairs $(s(K), \tau(K))$ for planar Minkowski-centered convex bodies is attained by determining lower and upper bounds on $\tau$ for each fixed $s$. Let $\varphi = (1+\sqrt{5})/2 \approx 1.618$, with $\hat s \approx 1.854$ as the unique solution of

$$\frac{(s^2+1)^2}{(s^2-1)\bigl(s^2+2s-1+2\sqrt{s(s^2-1)}\bigr)} = \frac{2\,(s^2-2s-1)}{(s-3)(s+1)}.$$

Then

$$\frac{2}{s+1} \leq \tau(K) \leq c(s) := \begin{cases} 1 & 1\leq s \leq \varphi \ \frac{(s^2+1)^2}{(s^2-1)(s^2+2s-1+2\sqrt{s(s^2-1)})} & \varphi < s \leq \hat s \ \frac{2\,(s^2-2s-1)}{(s-3)(s+1)} & \hat s < s \leq 2 \end{cases}$$

Every pair within these bounds is realized by some convex body $K$ (Dichter et al., 4 Dec 2025).

In the $(s, \tau)$-plane, the allowed region is the closed set bounded by $\tau = 2/(s+1)$ and $\tau = c(s),\ 1 \leq s \leq 2$. Below $s = \varphi$, these two curves coincide at $\tau=1$.

4. Diameter-Width Ratio: Asymmetry-Dependent Sharp Bound

For pseudo-complete $K$ with Minkowski center at the origin and $r(K,C)=1$, the diameter and width reduce to

$D(K,C) = 2(s(K)+1),$

$w(K,C) \geq \frac{2}{\tau(K)},$

yielding

$$\frac{D(K,C)}{w(K,C)} \leq \frac{(s(K)+1)\,\tau(K)}{2} \leq \frac{s(K)+1}{2} c(s(K)).$$

The global maximum occurs at $s=\varphi$, $\tau=1$, giving

$$\frac{D(K,C)}{w(K,C)} \leq \frac{\varphi+1}{2} \approx 1.309.$$

This is attained for the golden-house body

$$\mathbb{GH} = \mathrm{conv}\{\pm(1,0),\; \pm(1,-1),\; (0,\varphi)\}$$

with $C = K \cap (-K)$ (Dichter et al., 4 Dec 2025).

5. Special and Extremal Cases

The structure of extremal bodies and their parameter values organizes key boundary cases:

• Symmetric case ($s=1$): Centrally symmetric planar body; $\tau=1$, $D/w=1$.
• Golden house ($s=\varphi$): $\tau=1$; global maximum $D/w = (\varphi+1)/2$.
• Nearly symmetric ($1<s<\varphi$): Maximum diameter-width ratio for "almost symmetric" bodies, with $\tau\equiv1$.
• Intermediate regime ($\varphi < s < \hat s$): Maximal diameter-width ratio on the curve

$$\tau = \frac{(s^2+1)^2}{(s^2-1)\bigl(s^2+2s-1+2\sqrt{s(s^2-1)}\bigr)} < 1.$$

• Large asymmetry regime ($\hat s < s \leq 2$): Maximal ratio declines,

$\tau = \frac{2(s^2-2s-1)}{(s-3)(s+1)},$

from $\approx 0.78$ at $s \approx 1.854$ to $2/3$ at $s=2$ (triangle—the most asymmetric possible).

For each regime, explicit polygonal extremal bodies are constructed via support-line and homothety analysis (Dichter et al., 4 Dec 2025).

6. Comparison to Previous and Related Results

Earlier bounds for the diameter-width ratio in $\mathbb{R}^2$ were dimension-based and coarser. Brandenberg et al. proved the fundamental bound

$$\frac{D(K,C)}{w(K,C)} \leq \min\left\{\frac{s(K)+1}{2},\, \frac{s(K)^2}{s(K)^2-1}\right\},$$

obtaining a maximal ratio $\approx 1.42$ as opposed to Richter's earlier $D/w \leq 3$ (Brandenberg et al., 2023). In the Euclidean gauge $C=B_2$, the extremal "hood" construction yields a sharper bound

$$\frac{D(K,B_2)}{w(K,B_2)} \leq \frac{1}{2} (1+1/r) \approx 1.135$$

where $r\approx0.7935$ is the maximal inradius for pseudo-completeness in $B_2$ (Brandenberg et al., 2023).

7. Consequences and Duality Considerations

The asymmetry-sensitive results refine the classical $D/w \leq 3/2$ bound in the Euclidean plane and hold for any planar 0-symmetric gauge. The approach and feasible region for $\tau(K)$ transfer to dual parameters involving harmonic means; for example, the inclusion

$$((K^\circ-K^\circ)/2)^\circ \subset \gamma(K)\, \mathrm{conv}(K\cup(-K))$$

defines a dual feasible region for $\gamma(K)$ identically bounded as for $\tau(K)$ (Dichter et al., 4 Dec 2025).

The relation

$$\frac{D(K,C)}{w(K,C)} \leq \frac{s(K)+1}{2} c(s(K))$$

describes the exact, parameter-dependent tradeoff between diameter and width for planar pseudo-complete bodies and identifies explicit equality cases. The maximal diameter-width ratio is achieved exactly for bodies of the "golden house" type.

---

References:

• "Bounding the diameter-width ratio using containment inequalities of means of convex bodies" (Dichter et al., 4 Dec 2025)
• "From inequalities relating symmetrizations of convex bodies to the diameter-width ratio for complete and pseudo-complete convex sets" (Brandenberg et al., 2023)