Pentagons inscribed in a closed convex curve

Two theorems are proved. Let the points A₁, A₂, A₃, A₄, and A₅ be the vertices of a convex pentagon inscribed in an ellipse, let Κ⊂ℝ² be a convex figure, and let A₀ be a fixed distinguished point of its boundary ϖK. If the sum of any two of the neighboring angles of the pentagon A₁A₂A₃A₄A₅ is greater than π or the boundary ϖK is C⁴-smooth and has positive curvature, then some affine image of the pentagon A₁A₂A₃A₄A₅ is inscribed in K and has A₀ as the image of the vertex A₁. (This is not true for arbitrary pentagons incribed in an ellipse and for arbitrary convex figures.) Bibliography: 4 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 246, 1997, pp. 184–190. Translated by N. Yu. Netsvetaev.     

Minimal simplices inscribed in a convex body

Measuring how far a convex body $\mathcal{K }$ (of dimension $n$ ) with a base point ${O}\in \,\text{ int }\,\mathcal{K }$ is from an inscribed simplex $\Delta \ni {O}$ in “minimal” position, the interior point ${O}$ can display regular or singular behavior. If ${O}$ is a regular point then the $n+1$ chords emanating from the vertices of $\Delta $ and meeting at ${O}$ are affine diameters, chords ending in pairs of parallel hyperplanes supporting $\mathcal{K }$ . At a singular point ${O}$ the minimal simplex $\Delta $ degenerates. In general, singular points tend to cluster near the boundary of $\mathcal{K }$ . As connection to a number of difficult and unsolved problems about affine diameters shows, regular points are elusive, often non-existent. The first result of this paper uses Klee’s fundamental inequality for the critical ratio and the dimension of the critical set to obtain a general existence for regular points in a convex body with large distortion (Theorem A). This, in various specific settings, gives information about the structure of the set of regular and singular points (Theorem B). At the other extreme when regular points are in abundance, a detailed study of examples leads to the conjecture that the simplices are the only convex bodies with no singular points. The second and main result of this paper is to prove this conjecture in two different settings, when (1) $\mathcal{K }$ has a flat point on its boundary, or (2) $\mathcal{K }$ has $n$ isolated extremal points (Theorem C).     

Affine cross-polytopes inscribed in a convex body

Let X be an affine cross-polytope, i.e., the convex hull of n segments A ₁ B ₁,…, A _n B _n in mathbbRⁿ {mathbb{R}^n} that have a common midpoint O and do not lie in a hyperplane. The affine flag F(X) of X is the chain O ∈ L ₁ ⊂⋯ ⊂ L _n = mathbbRⁿ {mathbb{R}^n} , where L _k is the k-dimensional affine hull of the segments A ₁ B ₁,…, A _k B _k , k ≤ n. It is proved that each convex body K ⊂ mathbbRⁿ {mathbb{R}^n} is circumscribed about an affine cross-polytope X such that the flag F(X) satisfies the following condition for each k ∈{2,…, n}:the (k−1)-planes of support at A _k and B _k to the body L _k ∩ K in the k-plane L _k are parallel to L _k −1.Each such X has volume at least V(K)/2 ^n(n−1)/2. Bibliography: 5 titles.     

Closed polylines inscribed in a closed space curve

Let n be an odd positive integer. It is proved that if n + 2 is a power of a prime number and C is a regular closed non-self-intersecting curve in \mathbbRⁿ {\mathbb{R}^n} ,then C contains vertices of an equilateral (n + 2)-link polyline with n + 1 vertices lying in a hyperplane. It is also proved that if C is a rectifiable closed curve in \mathbbRⁿ {\mathbb{R}^n} ,then C contains n + 1 points that lie in a hyperplane and divide C into parts one of which is twice as long as each of the others. Bibliography: 6 titles.     

On polygons circumscribing a closed convex curve

Length and area formulas for closed polygonal curves are derived, as functions of the vertex angles and the distances to the lines containing the sides. Applications of the formulas are made to the class of polygons which circumscribe a given convex curve and have a prescribed sequence of vertex angles. Geometric conditions are given for polygons in the class which have extremal perimeter or area.     

On maximal convex lattice polygons inscribed in a plane convex set

Given a convex compact setK ? ?₂ what is the largestn such thatK contains a convex latticen-gon? We answer this question asymptotically. It turns out that the maximaln is related to the largest affine perimeter that a convex set contained inK can have. This, in turn, gives a new characterization ofK ₀, the convex set inK having maximal affine perimeter.     

Triangles inscribed in simple closed curves

We show that given any triangle T and any simple closed curve J ² there are infinitely many triangles similar to T whose vertices are contained in J. In fact, the set of such vertices is dense in J.     

Evolving a convex closed curve to another one via a length-preserving linear flow

Motivated by a recent curvature flow introduced by Professor S.-T. Yau [S.-T. Yau, Private communication on his “Curvature Difference Flow”, 2007], we use a simple curvature flow to evolve a convex closed curve to another one (under the assumption that both curves have the same length). We show that, under the evolution, the length is preserved and if the curvature is bounded above during the evolution, then an initial convex closed curve can be evolved to another given one.     

Rhombi inscribed in simple closed curves

We show that given any simple closed curveJ in ² and any lineL, the curveJ contains the four vertices of some rhombusR with two sides parallel toL. Furthermore, the cyclic order of the vertices ofR agrees with their cyclic order onJ. We also show that the diameters of the rhombi so produced (one for each lineL) may be bounded away from zero.     

Coverings by convex bodies and inscribed balls

Let be a Hilbert space. For a closed convex body denote by the supremum of the radiuses of balls contained in . We prove that for every covering of a convex closed body by a sequence of convex closed bodies , . It looks like this fact is new even for triangles in a 2-dimensional space.

     

On simplexes inscribed in the minor arc of a curve

On an arc L of length h, of class Cⁿ, and in Euclidean Eⁿ, the set of n+1 points (the partition of the arc) P={0, ₁ h, , _n–1h, h}, 0<₁<<_n–1<1 determines a simplex S_h(P) inscribed in the arc. For its volume V_h(P) we evaluate lim and prove that its greatest value is obtained for a unique choice of P=P_cr. The exact values for _i from P_cr are found for n=2, 3, 4, 5, 6.Translated from Ukrainskii Geometricheskii Sbornik, No. 33, pp. 114–120, 1990.     

Convergence on closed convex sets in a locally convex space

The purpose of this paper is to compare several kinds of convergences on the space C(X) of nonempty closed convex subsets of a locally convex space X. First we verify that the AW-convergence on C(X) is weaker than the metric Attouch-Wets convergence on C(X) of a metrizable locally convex space X. Moreover, we show that X is normable if and only if the two convergences on C(X × R) are equivalent. Secondly we define two convergences on C(X) analogous to the corresponding ones in a normed linear space, and investigate some basic properties of these convergences and compare them.     

Parallelepipeds and centrally symmetric hexagonal prisms circumscribed about a three-dimensional centrally symmetric convex body

Let K be a three-dimensional centrally symmetric compact convex set of unit volume. It is proved that K is contained in a centrally symmetric hexagonal prism (or a parallelepiped) of volume 4

Affine images of the rhombo-dodecahedron that are circumscribed about a three-dimensional convex body

The main result of the paper is dual to the author's earlier theorem on the affine images of the cube-octahedron (the convex hull of the midpoints of edges of a cube) inscribed in a three-dimensional convex body. The rhombododecahedron is the polyhedron dual to the cube-octahedron. Let us call a convex body in Κ⊂ℝ³ exceptional if it contains a parallelogram P and is contained in a cylinder with directrix P. It is proved that any nonexceptional convex body is inscribed in an affine image of the rhombo-dodecahedron. The author does not know if the assertion is true for all three-dimensional convex bodies. Bibliography: 2 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 246, 1997, pp. 191–195. Translated by N. Yu. Netsvetaev.     

Estimates for the minimal width of polytopes inscribed in convex bodies

The paper deals with the problem of approximating point sets byn-point subsets with respect to the minimal widthw. Let, in particular, ^d denote the family of all convex bodies in Euclideand-space, letA ^d and letn be an integer greater thand. Then we ask for the greatest number = _n (A) such that everyA A contains a polytope withn vertices which has minimal width at least w(A). We give bounds for _n ( ^d ), for _n (2133; ^d ), and for _n (W ^d ), where 2133; ^d ,W ^d denote the families of centrally symmetric convex bodies and of bodies of constant width, respectively.Dedicated to Professor L. danzer on the occasion of his sixtieth birthdayResearch for this paper was conducted in the academic year 1986/87 while both authors were visiting the University of Washington, Seattle. P. Gritzmann was supported by the Alexander von Humboldt Foundation.     

Example of an antiproximinal,but not a 2-antiproximinal convex closed bounded body.

An example of an antiproximinal but not 2-antiproximinal convex closed bounded body is constructed in the space c₀ endowed with Day’s norm.