Expectation of intrinsic volumes of random polytopes

Let K be a convex body in ℝ ^d , let j ∈ {1, …, d−1}, and let K(n) be the convex hull of n points chosen randomly, independently and uniformly from K. If ∂K is C ₊², then an asymptotic formula is known due to M. Reitzner (and due to I. Bárány if ∂K is C ₊³) for the difference of the jth intrinsic volume of K and the expectation of the jth intrinsic volume of K(n). We extend this formula to the case when the only condition on K is that a ball rolls freely inside K. Funded by the Marie-Curie Research Training Network “Phenomena in High-Dimensions” (MRTN-CT-2004-511953).     

On reduced convex bodies

A convex bodyK ⊂R ^d is called reduced if for each convex bodyK′ ⊂K,K′ ≠K, the width ofK′ is less than the width ofK. We prove that reduced bodyK is of constant width if (i) the bodyK has a supporting sphere almost everywhere in ∂K. (The radius of the sphere may vary with the point in ∂K; the condition (i) and strict convexity do not imply each other.) Supported by an N.S.E.R.C. Grant of Canada.     

Nerve complexes and moment-angle spaces of convex polytopes

We introduce spherical nerve complexes that are a far-reaching generalization of simplicial spheres, and consider the differential ring of simplicial complexes. We show that spherical nerve complexes form a subring of this ring, and define a homomorphism from the ring of polytopes to this subring that maps each polytope P to the nerve K _P of the cover of the boundary ∂P by facets. We develop a theory of nerve complexes and apply it to the moment-angle spaces Z _P of convex polytopes P. In the case of a polytope P with m facets, its moment-angle space Z _P is defined by the canonical embedding in the cone ℝ_≥ ^m . It is well-known that the space Z _P is homeomorphic to the polyhedral product (D ², S ¹)∂P* if the polytope P is simple. We show that the homotopy equivalence Z_P @ (D² ,S¹ )^K_P \mathcal{Z}_P \simeq (D^2 ,S^1 )^{K_P } holds in the general case. On the basis of bigraded Betti numbers of simplicial complexes, we construct a new class of combinatorial invariants of convex polytopes. These invariants take values in the ring of polynomials in two variables and are multiplicative with respect to the direct product or join of polytopes. We describe the relation between these invariants and the well-known f-polynomials of polytopes. We also present examples of convex polytopes whose flag numbers (in particular, f-polynomials) coincide, while the new invariants are different.     

Convex hull theorem for multiply connected domains in the plane with an estimate of the quasiconformal constant

For any multiply connected domain Ω in R2, let S be the boundary of the convex hull in H3 of R2\Ω which faces Ω. Suppose in addition that there exists a lower bound l > 0 of the hyperbolic lengths of closed geodesics in Ω. Then there is always a K-quasiconformal mapping from S to Ω, which extends continuously to the identity on S = Ω, where K depends only on l. We also give a numerical estimate of K by using the parameter l.     

Smooth convex bodies with proportional projection functions

For a convex body K ⊂ ℝⁿ and i ∈ {1, …, n − 1}, the function assigning to any i-dimensional subspace L of ℝⁿ, the i-dimensional volume of the orthogonal projection of K to L, is called the i-th projection function of K. Let K, K ₀ ⊂ ℝⁿ be smooth convex bodies with boundaries of class C ² and positive Gauss-Kronecker curvature and assume K ₀ is centrally symmetric. Excluding two exceptional cases, (i, j) = (1, n − 1) and (i, j) = (n − 2, n − 1), we prove that K and K ₀ are homothetic if their i-th and j-th projection functions are proportional. When K ₀ is a Euclidean ball this shows that a convex body with C ² boundary and positive Gauss-Kronecker with constant i-th and j-th projection functions is a Euclidean ball. The second author was supported in part by the European Network PHD, FP6 Marie Curie Actions, RTN, Contract MCRN-511953.     

The Bieberbach-Rademacher problem for the Monge-Ampère operator

Assume that Ω is a bounded, strictly convex, smooth domain in ℝ^N withN≥2. We consider the problem det ((∂ _iju(x)))=f(x,u(x)),u(x)→∞ asx→∂Ω, where (∂ _iju(x)) denotes the Hessian ofu(x) andf meets some natural regularity and growth conditions. We prove that there exists a unique smooth, strictly convex solution of this problem. The boundary-blow-up rate ofu(x) is characterized in terms of the distance ofx from ∂Ω. Partially supported by the Royal Swedish Academy of Sciences, Gustaf Sigurd Magnuson's fund.     

Test for properness using residue calculus

We will consider global problems in the ringK[X ₁, …,X _n] on the polynomials with coefficients in a subfieldK ofC. LetP=(P ₁, …,P _n):K ⁿ →K ⁿ be a polynomial map such that (P ₁,…,P _n) is a quasi-regular sequence generating a proper ideal, the main thing we do is to use the algebraic residues theory (as described in [5]) as a computational tool to give some result to test when a map (P ₁, …,P _n) is a proper map by computing a finite number of residue symbols.     

Polynomial hulls and an optimization problem

We say that a subset of Cⁿ is hypoconvex if its complement is the union of complex hyperplanes. We say it is strictly hypoconvex if it is smoothly bounded hypoconvex and at every point of the boundary the real Hessian of its defining function is positive definite on the complex tangent space at that point. Let B_n be the open unit ball in Cⁿ.Suppose K is a C^∞ compact manifold in ∂B₁ × Cⁿ, n > 1, diffeomorphic to ∂B₁ × ∂B_n, each of whose fibers K_z over ∂B₁ bounds a strictly hypoconvex connected open set. Let K be the polynomialhull of K. Then we show that K∖K is the union of graphs of analytic vector valued functions on B₁. This result shows that an unnatural assumption regarding the deformability of K in an earlier version of this result is unnecessary. Next, we study an H^∞ optimization problem. If pis a C^∞ real-valued function on ∂B₁× Cⁿ, we show that the infimum γ_ρ = inf_ƒ∈H ^∞ (B₁)ⁿ ‖ρ(z, ƒ (z))‖_∞ is attained by a unique bounded ƒ provided that the set (z, w) ∈ ∂B₁ × C ⁿ|ρ(z, w) ≤ γ_ρ has bounded connected strictly hypoconvex fibers over the circle.     

Analytic properties of conditional curvatures of convex hypersurfaces and the dirichlet problem for the Monge-Ampère equation

The existence and uniqueness of a surface with given geometric characteristics is one of the important topical problems of global differential geometry. By stating this problem in terms of analysis, we arrive at second-order elliptic and parabolic partial differential equations. In the present paper we consider generalized solutions of the Monge-Ampère equation ||z _ij || = ϕ(x, z, p) in Λ ⁿ , wherez = z(x ₁,...,z _n ) is a convex function,p = (p ₁,...,P _n) = (∂z/∂x ₁,...,ϖz/ϖx _n), andz _ij =ϖ ² z/ϖx _i ϖx _j. We consider the Cayley-Klein model of the space Λ ⁿ and use a method based on fixed point principle for Banach spaces. Translated fromMatematicheskie Zametki, Vol. 64, No. 5, pp. 763–768, November, 1998.     

Lower bounds at infinity of solutions of partial differential equations in the exterior of a proper cone

An extension of a classical theorem of Rellich to the exterior of a closed proper convex cone is proved: Let Γ be a closed convex proper cone inR ⁿ and −Γ′ be the antipodes of the dual cone of Γ. Let be a partial differential operator with constant coefficients inR ⁿ, whereQ(ζ)≠0 onR ⁿ−iΓ′ andP _i is an irreducible polynomial with real coefficients. Assume that the closure of each connected component of the set {ζ∈R ⁿ−iΓ′;P _j(ζ)=0, gradP _j(ζ)≠0} contains some real point on which gradP _j≠0 and gradP _j∉Γ∪(−Γ). LetC be an open cone inR ⁿ−Γ containing both normal directions at some such point, and intersecting each normal plane of every manifold contained in {ξ∈R ⁿ;P(ξ)=0}. Ifu∈ℒ′∩L _loc ² (R ⁿ−Γ) and the support ofP(−i∂/∂x)u is contained in Γ, then the condition implies that the support ofu is contained in Γ.     

Expectation of random polytopes

A random polytopeP _n in a convex bodyC is the convex hull ofn identically and independently distributed points inC. Its expectation is a convex body in the interior ofC. We study the deviation of the expectation ofP _n fromC asn→∞: while forC of classC ^k+1,k≥1, precise asymptotic expansions for the deviation exist, the behaviour of the deviation is extremely irregular for most convex bodiesC of classC ¹. Dedicated to my teacher and friend Professor Edmund Hlawka on the occasion of his 80^th birthday     

Estimating the diameter of the space of plane convex figures with respect to an affine-invariant metric

A convex figure K ⊂ ℝ² is a compact convex set with nonempty interior, and αK is a homothetic image of K with coefficient α ∈ ℝ. It is proved that for any two convex figures K₁, K₂ ⊂ ℝ² there is an affine transformation T of the plane such that K₁ ⊂ T(K₂) ⊂ 2.7K₁. Bibliography: 2 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 329, 2005, pp. 58–66.     

Quasiconformal and harmonic mappings between Jordan domains

Let Ω and Ω₁ be Jordan domains, let μ ∈ (0, 1], and let be a harmonic homeomorphism. The object of the paper is to prove the following results: (a) If f is q.c. and ∂Ω, ∂Ω₁ ∈ C ^1,μ , then f is Lipschitz; (b) if f is q.c., ∂Ω, ∂Ω₁ ∈ C ^1,μ and Ω₁ is convex, then f is bi-Lipschitz; and (c) if Ω is the unit disk, Ω₁ is convex, and ∂Ω₁ ∈ C ^1,μ , then f is quasiconformal if and only if its boundary function is bi-Lipschitz and the Hilbert transform of its derivative is in L ^∞. These extend the results of Pavlović (Ann. Acad. Sci. Fenn. 27:365–372, 2002).      

Blaschke's rolling theorem for manifolds¶with boundary

For a complete Riemannian manifold M with compact boundary ∂M denote by $\Cut$ the cut locus of $\f M$ in M. The rolling radius of M is roll(M)≔ dist(∂M, ?_{∂ M} ). Let Focal(∂M) be the focal distance of ∂M in M. Then conditions are given that imply the equality roll(M)= Focal(∂M). This generalizes Blaschke's rolling theorem from bounded convex domains in Euclidean space to more general Euclidean domains and to Riemannian manifolds with boundary. Received: 28 August 1998 / Revised version: 8 February 1999     

A Polynomial-Time Algorithm to Approximate the Mixed Volume within a Simply Exponential Factor

Let K=(K ₁,…,K _n ) be an n-tuple of convex compact subsets in the Euclidean space R ⁿ , and let V(⋅) be the Euclidean volume in R ⁿ . The Minkowski polynomial V _K is defined as V _K (λ ₁,…,λ _n )=V(λ ₁ K ₁+⋅⋅⋅+λ _n K _n ) and the mixed volume V(K ₁,…,K _n ) as

Hyperbolic manifolds with convex boundary

Let (M,∂M) be a 3-manifold, which carries a hyperbolic metric with convex boundary. We consider the hyperbolic metrics on M such that the boundary is smooth and strictly convex. We show that the induced metrics on the boundary are exactly the metrics with curvature K>-1, and that the third fundamental forms of ∂M are exactly the metrics with curvature K<1, for which the closed geodesics which are contractible in M have length L>2π. Each is obtained exactly once. Other related results describe existence and uniqueness properties for other boundary conditions, when the metric which is achieved on ∂M is a linear combination of the first, second and third fundamental forms.     

On the expected number of k-sets

Given a setS ofn points inR ^d , a subsetX of sized is called ak-simplex if the hyperplane aff(X) has exactlyk points on one side. We studyE _d (k,n), the expected number of k-simplices whenS is a random sample ofn points from a probability distributionP onR ^d . WhenP is spherically symmetric we prove thatE _d (k, n)≤cn ^d−1 WhenP is uniform on a convex bodyK⊂R ² we prove thatE ₂ (k, n) is asymptotically linear in the rangecn≤k≤n/2 and whenk is constant it is asymptotically the expected number of vertices on the convex hull ofS. Finally, we construct a distributionP onR ² for whichE ₂((n−2)/2,n) iscn logn. The authors express gratitude to the NSF DIMACS Center at Rutgers and Princeton. The research of I. Bárány was supported in part by Hungarian National Science Foundation Grants 1907 and 1909, and W. Steiger's research was supported in part by NSF Grants CCR-8902522 and CCR-9111491.     

On the area and perimeter of a random convex hull in a bounded convex set

Suppose K is a compact convex set in ℝ² and X _i , 1≤i≤n, is a random sample of points in the interior of K. Under general assumptions on K and the distribution of the X _i we study the asymptotic properties of certain statistics of the convex hull of the sample. Received: 24 July 1996/Revised version: 24 February 1998     

Average bending of convex pleated planes in hyperbolic three-space

If P is a pleated plane in 3-dimensional hyperbolic space H ³ and α a geodesic in its intrinsic metric we define B(P,α), the average bending of P in the direction α. We show that if P is a convex pleated plane embedded in H ³ then B(P,α)≤K for some universal K. Furthermore if P_Γ is a boundary component of the convex hull of a quasi-Fuchsian group Γ then B(P_Γ,α)=B(Γ) almost everywhere, where B(Γ) is a constant times the length of the bending lamination β_Γ of the pleated surface X _Γ=P_Γ/Γ. We use these to prove a number of results about quasi-Fuchsian groups including a universal bound on the Lipschitz constant for the map to infinity and a bound on the length of β_Γ by a constant times the Euler characteristic of X _Γ. Oblatum 10-X-1996 & 23-V-1997     

Characterization off-vectors of families of convex sets inR d Part I: Necessity of Eckhoff’s conditions

LetK=K ₁,...,K_n be a family ofn convex sets inR ^d . For 0≦i<n denote byf _i the number of subfamilies ofK of sizei+1 with non-empty intersection. The vectorf(K) is called thef-vectors ofK. In 1973 Eckhoff proposed a characterization of the set off-vectors of finite families of convex sets inR ^d by a system of inequalities. Here we prove the necessity of Eckhoff's inequalities. The proof uses exterior algebra techniques. We introduce a notion of generalized homology groups for simplicial complexes. These groups play a crucial role in the proof, and may be of some independent interest.