Minkowski Centered Convex Compact Sets

• Minkowski centered convex compact sets are convex bodies defined by having a designated center that guarantees optimal symmetric containment based on the Minkowski asymmetry.
• They underpin sharp geometric inequalities, including optimal harmonic-arithmetic mean relationships and explicit bounds on diameter-to-width ratios in the planar case.
• These sets are pivotal in convex geometry and functional analysis, as evidenced by extremal examples like the golden house pentagon which illustrate maximal asymmetry.

A Minkowski centered convex compact set is a fundamental object in convex geometry, encapsulating optimal symmetry properties, extremal containments, and key geometric functionals in finite dimensions—especially in the plane. Such sets play a central role in quantitative symmetrization inequalities, classification of extremal polytopes, and diameter-width ratio problems. The heart of the theory lies in the interplay among Minkowski centers, the Minkowski asymmetry, and precise comparison of symmetrizations, all admitting sharp, dimension-dependent thresholds and unique extremal examples such as the golden house pentagon.

1. Definitions: Minkowski Centered Sets and Asymmetry

Let $C$ be a full-dimensional compact convex set in $\mathbb{R}^d$. A point $c \in \mathbb{R}^d$ is called a Minkowski center of $C$ if

$C-c \subseteq -s(C) (C-c),$

where the Minkowski asymmetry $s(C)$ is

$$s(C) = \inf\{\lambda > 0 : \exists x \in \mathbb{R}^d \text{ such that } C-x \subseteq -\lambda(C-x)\}.$$

The set is Minkowski centered if $0$ is a Minkowski center, i.e.,

$C \subseteq -s(C) C.$

Key support-function characterization: $c$ is a Minkowski center iff for all $u \in \mathbb{R}^d$,

$h_{C-c}(u) \leq s(C) h_{C-c}(-u),$

with $h_C(u) = \max_{x \in C} \langle u, x \rangle$. In the planar case ($d=2$), the range $1 \leq s(C) \leq 2$ holds, with $s(C) = 1$ iff $C$ is centrally symmetric. Maximal asymmetry is achieved for simplices, which for $d=2$ is the triangle (Brandenberg et al., 2020).

2. Algebraic and Geometric Structure

Minkowski-centeredness is tightly linked to geometric and algebraic properties:

• Unique Center: When the balanced translate $C-c$ is both balanced and radially compact, the center is unique (Karn, 2022).
• Functional Analytic Duality: For compact convex sets with a Minkowski center, the space of continuous affine functions $A(C)$ is the dual of a base-normed space whose base has the given center.
• Gauge Interpretations: The Minkowski functional (gauge) $$\gamma_C(x) = \inf\{\lambda > 0 : x \in \lambda C\}$$ allows measurement of "balls" and circumcenters, leading to alternative intrinsic or gauge-centered notions of symmetry and minimal containment (Jahn, 2014).

3. Symmetrizations and Mean Sets

For any compact convex set $C$, two primary symmetrizations are considered:

• Arithmetic Mean: $\tfrac12(C + (-C))$, corresponding to the midpoint set.
• Harmonic Mean: $(C^\circ + (-C)^\circ)^\circ$, where $C^\circ$ is the polar of $C$.

Key chain of inclusions (planar case): $$C \wedge (-C) \subseteq C !- C \subseteq \tfrac12(C + (-C)) \subseteq \mathrm{conv}(C \cup (-C))$$ Here, $C !- C$ is the harmonic mean, and $C \wedge (-C)$ denotes the intersection. Simultaneous optimality for all inclusions occurs only for symmetric sets (Brandenberg et al., 2020).

4. Sharpened Containment Results: The Golden Ratio and Extremal Sets

A cornerstone result is the golden-ratio threshold for planar Minkowski-centered sets: $C\,!\,-C\;\copt\;\tfrac12\bigl(C+(-C)\bigr) \iff s(C) \leq \varphi = \tfrac{1+\sqrt5}{2} \approx 1.618$ where $\copt$ denotes optimal containment (no homothetic shrinking of the outer set possible).

• Extremal Example: The "golden house" pentagon is uniquely characterized (up to linear transformation) as the Minkowski-centered convex compact set for which $s(C) = \varphi$ and the above containment is optimal. Explicitly:

$$C_{\mathrm{GH}} = \mathrm{conv}\{ (-1,-1),\,(-1, 0),\,(0,\varphi),\,(1,0),\,(1,-1) \}$$

This pentagon realizes the maximal asymmetry compatible with optimal harmonic-arithmetic mean containment (Brandenberg et al., 2020).

• Generalization to Other Means: Analogous sharp inequalities hold for containment of $K \cap (-K)$ in $\frac{K-K}{2}$, with bounds on the containment factor $\tau(K)$ as functions of $s(K)$, e.g.,

$\frac{2}{s(K)+1} \leq \tau(K) \leq 1,$

with explicit piecewise formulae in the planar case (Dichter et al., 4 Dec 2025).

5. Extremal Inequalities and Diameter-Width Ratios

The structure of Minkowski-centered convex compact sets allows the derivation of sharp diameter-to-width ratio bounds for pseudo-complete sets. Given a symmetric gauge body $C$ and pseudo-complete $K$: $$\frac{D(K,C)}{w(K,C)} \leq \frac{s(K)+1}{2}\, c(s(K))$$ where $c(s)$ is an explicit function piecewise defined in terms of $s(K)$, with the maximum $\frac{\varphi+1}{2} \approx 1.309$ attained by the golden house (Dichter et al., 4 Dec 2025, Brandenberg et al., 2023). For $s(K) = 2$ (triangle), this ratio reaches its minimal sharp value.

6. Geometric and Combinatorial Properties

For planar Minkowski-centered convex sets with $s(K) > \varphi$, the intersection $\operatorname{bd}(K) \cap \operatorname{bd}(-K)$ consists of exactly six points—a result connected with the combinatorics of optimal support, and mirrored in the geometric transition from the golden house (five points) to triangles (six points at maximal asymmetry) (Brandenberg et al., 2023). These intersection patterns are crucial in the classification of extremal polytopes for symmetrization inequalities.

7. Connections and Applications

Minkowski centered convex compact sets bridge convex geometry, functional analysis, and optimization. Their properties underlie:

• Classification of affine extremal bodies for symmetrization and containment inequalities (Brandenberg et al., 2020).
• Explicit construction of extremal and interpolating polytopes realizing all possible pairs $(s, \tau)$ in the plane (Dichter et al., 4 Dec 2025).
• Applications to functional-analytic dualities and the structure of affine function spaces on compact convex sets (Karn, 2022).
• Geometric realization of bounds for ratios such as diameter/width, illuminating the fundamental role of asymmetry in geometric optimization (Brandenberg et al., 2023).

The Minkowski centered convex compact set thus represents an archetype for the study of asymmetry, extremality, and optimal containment in convex geometry, with deep ramifications in analysis and metric geometry.