Ellipsoids of maximal volume in convex bodies

The largest discs contained in a regular tetrahedron lie in its faces. The proof is closely related to the theorem of Fritz John characterizing ellipsoids of maximal volume contained in convex bodies.     

Approximation of convex bodies by axially symmetric bodies

Let be an arbitrary planar convex body. We prove that contains an axially symmetric convex body of area at least . Also approximation by some specific axially symmetric bodies is considered. In particular, we can inscribe a rhombus of area at least in , and we can circumscribe a homothetic rhombus of area at most about . The homothety ratio is at most . Those factors and , as well as the ratio , cannot be improved.

     

Centrally symmetric convex bodies and distributions

To each centrally symmetric convex body is assigned a distribution on the sphere. As applications, geometric formulas and a characterization of zonoids are obtained.     

Some problems concerning centrally symmetric convex bodies

The conjecture that among convex bodies Q in Rⁿ, with a center of symmetry at the origin, for which , the value of is a maximum when Q is the layer between two hyperplanes, is proved for n=2 and n=3. Various approaches to the problem are discussed as well as related unsolved problems. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akad. Nauk SSSR, Vol. 45, pp. 75–82, 1974.     

Centrally symmetric convex bodies and distributions II

In continuation of a previous work we study the generating distributions of centrally symmetric convex bodies and obtain some more geometric formulas and new characterizations of zonoids and generalized zonoids.     

Plane sections of centrally symmetric convex bodies

The note contains an example of three plane convex centrally symmetric figuresP ₁,P ₂,P ₃ such that no centrally symmetric 3-dimensional body has three coaxial central affinely equivalent toP ₁,P ₂,P ₃ respectively.     

Homothetic packings of centrally symmetric convex bodies

Geometriae Dedicata - We estimate the bottom of the $$L^2$$ spectrum of the Laplacian on locally symmetric spaces in terms of the critical exponents of appropriate Poincaré series. Our main...     

A Blichfeldt-type inequality for centrally symmetric convex bodies

In this note, we derive an asymptotically sharp upper bound on the number of lattice points in terms of the volume of a centrally symmetric convex body. Our main tool is a generalization of a result of Davenport that bounds the number of lattice points in terms of volumes of suitable projections.     

Nearly all convex bodies are smooth and strictly convex

By an old result of Klee, those convex bodies which are not smooth or not strictly convex form a set of first Baire category. It is proved here that they are “even fewer”: they only form a σ-porous set.     

Asymptotic formulas for the diameter of sections of symmetric convex bodies

Sharpening work of the first two authors, for every proportion λ∈(0,1) we provide exact quantitative relations between global parameters of n-dimensional symmetric convex bodies and the diameter of their random ⌊λn⌋-dimensional sections. Using recent results of Gromov and Vershynin, we obtain an “asymptotic formula” for the diameter of random proportional sections.     

Blaschke-decomposable convex bodies

For a given centred convex bodyK of ℝ,n≥3, let be the class of all convex bodies with the same projection body asK. The question whetherK can be expressed as a Blaschke average of two non-homothetic bodies from is considered. Necessary and sufficient conditions onK to be Blaschke decomposable in are given. The paper provides also a characterization of the bodiesK such that the Blaschke indecomposable bodies in are dense in itself.