Strictly Convex Normed Planes

• Strictly convex normed planes are two-dimensional spaces where the unit ball has a boundary free of flat segments, ensuring that every chord between distinct points lies within the open ball.
• This property guarantees unique supporting lines, bisector regularity, and a one-to-one correspondence between ball hull arcs and intersections, which simplifies geometric analysis.
• Algorithmically, strict convexity enables efficient constructions for ball hulls and intersections, and it underpins the rigidity and linearity of distance-preserving isometries in these spaces.

A strictly convex normed plane is a two-dimensional real normed vector space whose closed unit ball possesses strict convexity: for every pair of distinct unit vectors, the line segment joining them lies entirely in the open unit ball except at the endpoints. This geometric property excludes flat pieces on the boundary of the unit ball and underlies a variety of uniqueness and regularity results for convex sets, bisectors, isometries, and optimization phenomena in the plane. The study of strictly convex normed planes spans constructive convexity theory, algorithmic geometry, variational calculus, and the structure theory of isometries and metric spaces.

1. Definition and Geometric Structure

Let $(X, \|\cdot\|)$ be a real normed vector space of dimension two. The closed unit ball is $B = \{x \in X : \|x\| \leq 1\}$, with boundary $\partial B = \{x \in X : \|x\| = 1\}$, called the unit circle or unit sphere. The space $X$ is strictly convex if for all distinct $x, y \in \partial B$,

$\left\| \frac{x + y}{2} \right\| < 1.$

Equivalently, no nontrivial segment lies on $\partial B$. In the planar case, the unit circle is a compact, centrally symmetric, convex curve whose tangent direction changes strictly monotonically as one moves along the boundary; there are no straight-edge segments and no corners. For any point $x \in \partial B$, there is a unique supporting line not crossing the interior. This ensures that the boundary is a strictly convex, simple, closed curve, but not necessarily differentiable everywhere (Mandelkern, 2024, Sánchez et al., 2024).

Examples:

• Euclidean norm: $\|(x, y)\|_2 = \sqrt{x^2 + y^2}$, whose unit circle is the round circle—smooth, strictly convex.
• $\ell^p$ norms for $1 < p < \infty$, $p \neq 2$: The unit circle is a "superellipse"—strictly convex, but the curvature varies.
• Polytopal/unit balls with flat segments (as for the $\ell^1$ and $\ell^\infty$ norms) are not strictly convex.

Every closed, centrally symmetric, strictly convex plane curve can be used as the unit circle for a strictly convex norm (Martín et al., 2014).

2. Characterizations and Analytic Criteria

Strict convexity admits several equivalent characterizations in two dimensions (Mandelkern, 2024, Sánchez et al., 2024):

• No boundary segments: $\partial B$ contains no straight line segment of positive length.
• Midpoint strict inequality: For $\|x\| = \|y\| = 1$, $x \neq y$, $\left\| \tfrac{x + y}{2} \right\| < 1$.
• Extreme points: Every boundary point of $B$ is an extreme point.
• Norm additivity: If $\|x + y\| = \|x\| + \|y\|$ and $\|x+y\| \neq 0$, then $x$ and $y$ are linearly dependent.
• Bisector uniqueness: Every nonzero $z \in X$ belongs to exactly one bisector $B(-u, u)$, up to sign; equivalently, the equation $\|z-u\| = \|z+u\|$, $\|u\|=1$, has exactly two solutions $u, -u$ (Sánchez et al., 2024).

In terms of ball geometry, strict convexity is equivalent to the uniqueness of intersection points of osculating disks; in strictly convex planes, any two osculating balls (with centers and radii such that their boundaries "kiss") intersect in at most one point. Constructively, strict convexity ensures uniqueness (but not necessarily existence) of such osculating points (Mandelkern, 2024).

3. Metric Geometry: Bisectors, Ball Hulls, and Algorithms

In strictly convex normed planes, metric objects exhibit strong regularity:

• Bisectors: The locus $B(a, b) = \{ z \in X : \|z - a\| = \|z - b\| \}$ is, for $a \neq b$, a homeomorphic image of $\mathbb{R}$, i.e., a simple unbroken curve with two arms diverging to infinity. There are no segments in bisectors and every such bisector is determined uniquely by the endpoints (Sánchez et al., 2024).
• Ball Hull and Ball Intersection: The $\lambda$-ball hull $\bh(K, \lambda)$ of a finite set $K$ is the intersection of all balls of radius $\lambda$ containing $K$, while the ball intersection $\bi(K, \lambda)$ is the intersection of balls centered at the points of $K$ with radius $\lambda$. In strictly convex planes, the boundary of the ball hull is formed by minimal circular arcs joining pairs of points of $K$, and there is a duality: each vertex of the ball intersection corresponds to an arc in the ball hull and vice versa (Martín et al., 2014, Martín et al., 2014).

Algorithmic Implications

Strict convexity allows the extension of Euclidean geometric algorithms to the normed setting:

• Construction of ball hulls and intersections is achievable in $O(n \log n)$ time for $n$ points.
• The 2-center problem (finding two disks of given radii covering all points in $K$) admits deterministic algorithms of complexity $O(n^2)$ to $O(n^2 \log n)$.
• Geometric predicates, such as inclusion or separation, behave analogously to the Euclidean setting due to the uniqueness of arcs and intersection points, making the tools robust under changes of norm so long as strict convexity is preserved (Martín et al., 2014).

4. Isometries, Affine Geometry, and Rigidity

A fundamental consequence of strict convexity is that the group of distance-preserving transformations in the plane is sharply restricted:

• If a surjection $\phi: X \to X$ preserves unit distance (i.e., $\|x-y\|=1 \implies \|\phi(x) - \phi(y)\|=1$), then in strictly convex normed planes, $\phi$ is necessarily an affine isometry (Gehér, 2015). This solves the Aleksandrov conservative distance problem in the affirmative for strictly convex norms.
• The absence of flat segments is essential: if the unit circle has flat segments of length exceeding 1, non-affine distance-preserving transformations can exist. Strict convexity ensures the unique construction of "equilateral triangles" on any base and the global extension property for isometries.
• Any isometry between the unit spheres of strictly convex normed planes is linear provided non-differentiability points exist and the spheres are piecewise differentiable. The result extends to isometries between arbitrary piecewise $C^1$ Jordan convex curves provided both spaces have finite—greater than two—non-smooth directions (Sánchez, 2021).

5. Convex Analysis, Homogenization, and the Stable Norm

In variational and homogenization settings, strictly convex normed planes arise naturally:

• The stable norm $\phi$ on $\mathbb{R}^2$, defined via periodic perimeter homogenization, is always strictly convex in the sense that $\phi^2$ is strictly convex (Chambolle et al., 2012).
• Differentiability of the stable norm at a vector $p \neq 0$ is equivalent to the associated family of plane-like minimizers foliating the plane; totally irrational directions always admit differentiability, while rational directions may manifest corners (non-differentiable points) but strictly convex boundaries (no flat segments).
• The absence of flat segments in the unit ball of the stable norm underpins regularity results for minimizers in geometric variational problems.

6. Constructive Aspects and Limitations

From a constructive mathematics perspective, strict convexity retains its characterizations regarding uniqueness but not necessarily existence:

• Uniqueness of osculation points—the intersection of osculating balls—is guaranteed under strict convexity, but classically equivalent existence statements translate constructively into requirements for metric convexity or completeness (Mandelkern, 2024).
• Classical theorems relying on the guarantee of intersection points (such as the existence of a unique touchpoint for osculating balls) may fail constructively; only the uniqueness component is inherited by strict convexity without further metric assumptions.

7. Further Theoretical Implications and Open Directions

Strict convexity in two dimensions provides a setting where many classical Euclidean structures generalize with full rigor. However, notable phenomena and subtleties arise:

• The characterization of Euclidean planes among strictly convex normed planes depends on the global shape of bisectors: only if every central symmetric bisector is a straight line (full rotational symmetry) do we obtain a Euclidean norm (Sánchez et al., 2024).
• Extending geometric algorithms (e.g., for hulls or covering sets) to non-strictly convex (e.g., polyhedral) norms is a major open direction. In non-strictly convex planes, ball hull and intersection boundaries may involve segments, and algorithmic schemes require deeper case analysis to account for such degeneracies (Martín et al., 2014).
• In metric analysis and convex geometry, differentiability of the norm, duality between primal and dual spheres, and isometric rigidity continue to motivate foundational investigations in the theory of normed vector spaces and their applications in geometric analysis.

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References:

• (Mandelkern, 2024) "Convexity and Osculation in Normed Spaces"
• (Martín et al., 2014) "The 2-center problem and ball operators in strictly convex normed planes"
• (Gehér, 2015) "A contribution to the Aleksandrov conservative distance problem in two dimensions"
• (Sánchez et al., 2024) "A characterisation of Euclidean normed planes via bisectors"
• (Martín et al., 2014) "Algorithms for ball hulls and ball intersections in strictly convex normed planes"
• (Chambolle et al., 2012) "Plane-like minimizers and differentiability of the stable norm"
• (Sánchez, 2021) "Linearity of isometries between convex Jordan curves"