Let K be a convex compact body with nonempty interior in the d-dimensional Euclidean space Rd and let x1, …, xn be random points in K, independently and uniformly distributed. Define Kn = conv {x1, …, xn}. Our main concern in this paper will be the behaviour of the deviation of vol Kn from vol K as a function of n, more precisely, the expectation of the random variable vol (K\Kn). We denote this expectation by E (K, n).

A geometric property of convex sets which is equivalent to a minimax inequality of the Ky Fan type is formulated. This property is used directly to prove minimax inequalities of the von Neumann type, minimax inequalities of the Fan-Kneser type, and fixed point theorems for inward and outward maps.

We construct a polyhedron with ten vertices of genus three which has three axes of symmetry. It is as symmetric as possible. Ten is the minimal number of vertices which a polyhedron of genus three can have. A modification of our polyhedron yields a symmetric polyhedral realization of Dyck's regular map.

We construct realizations of Dyck's regular map of genus three as polyhedra in ℝ3. One of these has one axis of symmetry of order three and three axes of symmetry of order two. The other polyhedra have three axes of symmetry. We show that a polyhedron realizing Dyck's regular map cannot have a symmetry group of order larger than six. Thus the symmetry groups of our realizations are maximal.

There exists a natural notion of convexity in the space of all compact convex sets in D. Thus, we may consider the space of all bounded closed convex families of compact convex sets. We present here a strange generic extremal behaviour of the elements of this space.

For any two compact convex sets in a Euclidean space, the relation between the volume of the sum of the two sets and the volume of each of them is given by the Brünn-Minkowski inequality. In this note we prove an analogous relation for the one-dimensional Hausdorff measure of the one-skeleton of the above sets. Also, some counterexamples are given which show that the above results are the best possible in some special cases.

It is shown that for arbitrary ε > 0 there is a function x(t, x) defined on the square [0,1] × [0,1] such that x(t, s) represents an extremal point of the unit ball in the space of Lipschitz continuous functions, and the gradient of x(t, s) is equal to 0 except on a set of measure at most ε.

We show that the number of combinatorially distinct labelled d-polytopes on n vertices is at most , as n/d → ∞. A similar bound for the number of simplicial polytopes has previously been proved by Goodman and Pollack. This bound improves considerably the previous known bounds. We also obtain sharp upper and lower bounds for the numbers of real oriented and unoriented matroids with n elements of rank d. Our main tool is a theorem of Milnor and Thorn from real algebraic geometry.

We show that if a polytope K1, in ℝd can be partitioned into a finite number of sets, and these sets can be moved by isometries in a locally discrete group to form a convex body K2, then K2 is a polytope and a similar partition can be made where the sets involved are simplices with disjoint interiors. This gives partial answers to questions of Tarski, Sallee and Wagon.

The principal objective of this work is to investigate various classes of centrally symmetric convex sets. These classes range from the zonoids at one extreme to the class of all centrally symmetric bodies at the other. The defining properties of these classes involve inequalities between mixed volumes. Various other characterizations will be found in response to a number of questions in a recent survey article by Rolf Schneider and Wolfgang Weil. Some of these are concerned with measures on a Grassmannian manifold while others relate to the intermediate surface area measures of convex bodies. We shall also show these classes are characterized by certain extremal geometric inequalities. The work concludes with a brief discussion of related results concerned with generalized zonoids.

It is shown that every compact convex set in with mean width equal to that of a line segment of length 2 and with Steiner point at the origin is contained in the unit ball. As a consequence, the diameter with respect to the Hausdorff metric of the space of all such sets is 1. There also results a sharp bound for the Hausdorff distance between any two compact convex sets.

We describe a toroidal polyhedral map which can be geometrically realized in R3 but not via a Schlegel diagram of a convex 4-polytope. Moreover, this map is not isomorphic to a subcomplex of the boundary complex of any convex polytope.

If C and Co are two convex bodies in Ed we say that C slides (rolls) freely inside Co if the following condition is satisfied: for each x ∈ ∂C0 (and each rotation R) there is a translation t such that, if gC = C + t (= RC + t), then gC ⊂ Co and x ∈ ∂gC. This work establishes certain topological conditions which ensure the free rolling and sliding of C inside Co. One consequence of these conditions is that, if ∂K ∩ int gK is a topological ball for all rigid motions g, then K is a ball in the geometrical sense.

Let be k non-overlapping translates of the unit d-ball Bd in euclidean d-space Ed. Let Ck denote the convex hull of their centres and let Sk be a segment of length 2(k– 1). Furthermore, let Vd denote the d-volume. L. Fejes Tóth conjectured in [1], that, for d ≥ 5,

We prove in this note that if two convex curves are internally tangent (lie on the same side of the common tangent), then in most cases they also cut each other infinitely many times. It follows that, in a certain sense, if two convex curves have an odd number of common points, then in general they are externally tangent (lie on different sides of the common tangent). The sense of the expressions most cases and in general remains to be precised.

Let S be a simplex in En which is homothetic to a given simplex S*, which contains no point of the integral lattice in its interior, and which has maximal volume V(S). We conjecture that V(S) > nn/n!, and establish the conjecture for n < 3.

Semiconvex sets are objects in the algebraic variety generated by convex subsets of real linear spaces. It is shown that the fundamental notions of convex geometry may be derived from an entirely algebraic approach, and that conceptual advantages result from applying notions derived from algebra, such as ideals, to convex sets. Some structural decomposition results for semiconvex sets are obtained. An algebraic proof of the algebraic Hahn-Banach theorem is presented.

The purpose of this paper is to prove two integralgeometric formulae for convex bodies. Our results are expressed in terms of integrals with respect to the rigid-motion-invariant measure μd, r on the space ℰ(d, r) of all r-dimensional affine flats in d-dimensional Euclidean space Ed. Rolf Schneider, in an unpublished note [6], has shown that for a convex polytope P in Ed and 1 ≤ r ≤ d – 1 one has

where ηr(P) is the sum of the contents of the r-dimensional faces of P, ηo(Ed–r ∩ P) is the number of vertices of the (d – r)-dimensional section Ed–r ∩ P, and α(r) is the content of the r-dimensional unit ball.

Let K be a convex body in Ed and let skelsK denote the s-skeleton of K. Let ηs(K) denote the Hausdorff s-measure of skelsK and

Recently the notion of elementary symmetrization attracted new attention in the field of convex sets (see [L; W]), and it was proved that Minkowski's “Quermassintegrale” are decreased by elementary symmetrization. On the other hand, the concept of Schwarz symmetrization for Borel functions gained new interest from its possible applications in the field of elliptic partial differential equations (see [S1, S2, HI, T1, T2, HY1, HY2, GL1, GL2]).