---
title: Diameter-Width Ratio in Planar Pseudo-Complete Sets
url: https://www.emergentmind.com/topics/diameter-width-ratio-for-planar-pseudo-complete-sets
type: topic
---

# Diameter-Width Ratio in Planar Pseudo-Complete Sets

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 $K$ [2512.04633].

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 $C = K \cap (-K)$ [2512.04633].

## 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 [2512.04633].

## 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 $D/w \leq 3$ [2306.11460]. 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 $B_2$ [2306.11460].

## 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 $\tau(K)$ [2512.04633].

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" [2512.04633]  
- "From inequalities relating symmetrizations of convex bodies to the diameter-width ratio for complete and pseudo-complete convex sets" [2306.11460]

Source: https://www.emergentmind.com/topics/diameter-width-ratio-for-planar-pseudo-complete-sets