On Reuleaux triangles in Minkowski planes.
On Separating Two Simple Polygons by a Single Translation.
On seven points in the boundary of a plane convex body in large relative distances.
On the affine convexity of convex curves and hypersurfaces.
On the area sum of a convex polygon and its polar reciprocal.
On the area sum of a convex set and its polar reciprocal.
On the Exact Constant in the Quantitative Steinitz Theorem in the Plane.
On the Graph of Large Distances.
On the Hadwiger numbers of centrally symmetric starlike disks.
On the Minimal Convex Annulus of a Planar Convex Body.
On the Non-Existence of Certain Nearly Regular Planar Maps with Two Exceptional Faces
On the Sobolev distance of convex bodies.
On the stability of the unit circle with minimal self-perimeter in normed planes
We prove a stability result on the minimal self-perimeter L(B) of the unit disk B of a normed plane: if L(B) = 6 + ε for a sufficiently small ε, then there exists an affinely regular hexagon S such that S ⊂ B ⊂ (1 + 6∛ε) S.
On the Union of Jordan Regions and Collision-Free Translational Motion Amidst Polygonal Obstacles.
Optimizing geometric measures for fixed minimal annulus and inradius.
Packing and covering a unit equilateral triangle with equilateral triangles.
Pairs of convex bodies in a hyperspace over a Minkowski two-dimensional space joined by a unique metric segment
Let be a Minkowski space with a unit ball and let be the Hausdorff metric induced by in the hyperspace of convex bodies (nonempty, compact, convex subsets of ℝ). R. Schneider [RSP] characterized pairs of elements of which can be joined by unique metric segments with respect to for the Euclidean unit ball Bⁿ. We extend Schneider’s theorem to the hyperspace over any two-dimensional Minkowski space.
Parallelograms inscribed in a curve having a circle as π/2-isoptic
Jean-Marc Richard observed in [7] that maximal perimeter of a parallelogram inscribed in a given ellipse can be realized by a parallelogram with one vertex at any prescribed point of ellipse. Alain Connes and Don Zagier gave in [4] probably the most elementary proof of this property of ellipse. Another proof can be found in [1]. In this note we prove that closed, convex curves having circles as π/2-isoptics have the similar property.
Partial inflation of closed polygons in the plane.
Piecewise Convex Curves and their Integral Representation
2000 Mathematics Subject Classification: 52A10.A convex arc in the plane is introduced as an oriented arc G satisfying the following condition: For any three of its points c1 < c2 < c3 the triangle c1c2c3 is counter-clockwise oriented. It is proved that each such arc G is a closed and connected subset of the boundary of the set FG being the convex hull of G. It is shown that the convex arcs are rectifyable and admit a representation in the natural parameter by the Riemann-Stieltjes integral...