The geometry of Minkowski spaces -- a survey. Part I

We survey elementary results in Minkowski spaces (i.e. finite dimensional Banach spaces) that deserve to be collected together, and give simple proofs for some of them. We place special emphasis on planar results. Many of these results have often been rediscovered as lemmas to other results. In Part I we cover the following topics: The triangle inequality and consequences such as the monotonicity lemma, geometric characterizations of strict convexity, normality (Birkhoff orthogonality), conjugate diameters and Radon curves, equilateral triangles and the affine regular hexagon construction, equilateral sets, circles: intersection, circumscribed, characterizations, circumference and area, inscribed equilateral polygons.

💡 Research Summary

The paper presents a comprehensive survey of elementary geometric results in finite‑dimensional Banach spaces, commonly called Minkowski spaces, with a special focus on the planar case. Its purpose is to gather scattered lemmas and simple observations that have repeatedly resurfaced in the literature, to provide unified statements, and to supply concise proofs that highlight the underlying geometric intuition.

The exposition begins with the triangle inequality, the cornerstone of any normed space. From this basic estimate the author derives the monotonicity lemma, which asserts that the distance between two points varies monotonically along the line segment joining them. This lemma becomes a versatile tool for later arguments concerning convexity and orthogonality.

Next, the paper investigates strict convexity. Several equivalent characterizations are given: (i) every boundary point is a non‑extreme point of the unit ball, (ii) the unit sphere contains no non‑trivial line segments, and (iii) the norm is Fréchet differentiable away from the origin. These conditions are linked to Birkhoff orthogonality (also called normality). In a Minkowski plane a vector (x) is Birkhoff‑orthogonal to (y) if (|x| \le |x+\lambda y|) for all real (\lambda). The author shows that this asymmetric notion coincides with the usual inner‑product orthogonality only in the Euclidean case, and that in two dimensions every direction admits a unique Birkhoff‑orthogonal direction.

The discussion then moves to conjugate diameters and Radon curves. A pair of diameters of the unit ball is called conjugate if each is Birkhoff‑orthogonal to the other. The paper proves that every Minkowski plane possesses at least one such pair, and that the boundary curve determined by the unit ball can be chosen to be a Radon curve— a centrally symmetric, strictly convex curve for which all conjugate diameters are mutually orthogonal. This provides a geometric analogue of orthogonal bases in Euclidean geometry.

The author proceeds to the construction of equilateral triangles and the affine regular hexagon. Starting from any two points, one can locate a third point at the same norm‑distance from each, thereby forming an equilateral triangle. Repeating this process yields a regular hexagon that is affine‑equivalent to the Euclidean regular hexagon, regardless of the underlying norm. This construction illustrates how affine transformations preserve the combinatorial structure of equilateral configurations even when metric properties change.

Equilateral sets are examined next. In an (n)-dimensional Minkowski space the maximal cardinality of a set of pairwise equidistant points is (n+1). The paper supplies a short proof based on the linear independence of the vectors joining a fixed point to the others, and discusses the sharpness of this bound by exhibiting explicit examples in low dimensions.

The bulk of the survey is devoted to circles. The author treats three classical problems: (1) the maximal number of intersection points of two Minkowski circles (which is at most two, as in the Euclidean case), (2) conditions for the existence and uniqueness of a circumscribed circle around a given triangle, and (3) various characterizations of circles via constant width, support functions, and symmetry. The paper also derives a Minkowski analogue of the classical relationship between circumference and area, showing that the ratio depends on the shape of the unit ball and reduces to (\pi) only in the Euclidean norm.

Finally, the paper investigates inscribed equilateral polygons. It proves that an equilateral (k)-gon can be inscribed in a Minkowski circle if and only if the unit ball admits a rotational symmetry of order (k). Consequently, regular hexagons always exist (as shown earlier), while regular pentagons may fail to be inscribable unless the norm possesses the appropriate symmetry.

Throughout the survey, the author emphasizes that many of these results have been rediscovered as auxiliary lemmas in unrelated research areas, such as functional analysis, convex geometry, and optimization. By presenting them together with streamlined proofs, the paper not only clarifies the logical dependencies among the various concepts but also provides a ready reference for researchers who need reliable geometric tools in Minkowski spaces. The systematic treatment of planar phenomena, together with occasional remarks on higher‑dimensional extensions, makes the article a valuable bridge between elementary convex geometry and more advanced Banach‑space theory.