We give a geometric characterization of inner product spaces among all finite dimensional real Banach spaces via concurrent chords of their spheres. Namely, let x be an arbitrary interior point of a ball of a finite dimensional normed linear space X. If the locus of the midpoints of all chords of that ball passing through x is a homothetical copy of the unit sphere of X, then the space X is Euclidean. Two further characterizations of the Euclidean case are given by considering parallel chords of 2-sections through the midpoints of balls.
We prove that, unexpectedly, the illumination number of a direct vector sum of convex bodies is not necessarily equal to the product of the illumination numbers of its summands, and we describe a condition under which equality holds. Some further results and two research problems are presented.
Let P ⊂ Rn be a centrally symmetric, convex n-polytope with 2r vertices, n ≥ 2. Let P be a family of m ≥ n + 1 homothetical copies of P. Based on an algorithmical approach to center hyperplanes of finite point sets in Minkowski spaces with polyhedral norms,
we show that a hyperplane transversal of all members of P (if it exists) can be found in O(rm) time when the dimension n is fixed.
The purpose of this paper is to establish extremal values for inner and
outer radii of the unit ball of a Minkowski space for the Holmes--Thompson and
Busemann measures. Furthermore, we confirm a conjecture of C. A. Rogers and G. C. Shephard on ellipsoids.
Napoleon's original theorem refers to arbitrary triangles in the
Euclidean plane. If equilateral triangles are externally erected on the sides
of a given triangle, then their three corresponding circumcenters form an
equilateral triangle. We present some analogous theorems and related statements
for the isotropic (Galilean) plane.
A convex body K in ℝd is said to be reduced if the minimum width of each convex body properly contained in K is strictly smaller than the minimum width of K. We study the question of Lassak on the existence of reduced polytopes of dimension larger than two. We show that a pyramid
of dimension larger than two with equal numbers of facets and vertices is not reduced. This generalizes the main result from
Some theorems from inversive and Euclidean circle geometry are extended to all affine Cayley-Klein planes. In particular,
we obtain an analogue to the first step of Clifford’s chain of theorems, a statement related to Napoleon’s theorem, extensions
of Wood’s theorem on similar-perspective triangles and of the known fact that the three radical axes of three given circles
are parallel or have a point in common. For proving these statements, we use generalized complex numbers.
Studying the relation between the length of a chord of the unit circle and the length of the arc corresponding to it, some
new characterizations of the Euclidean plane among all normed planes are obtained. All these results yield characterizations
of inner product spaces in higher dimensions.