Let be a finite-dimensional vector space over a square-root closed ordered field (this restriction permits an inner product with corresponding norm to be imposed on ). Many properties of the family :=() of convex polytopes in can be expressed in terms of valuations (or finitely additive measures). Valuations such as volume, surface area and the Euler characteristic are translation invariant, but others, such as the moment vector and inertia tensor, display a polynomial behaviour under translation. The common framework for such valuations is the polytope (or Minkowski) ring Π:=Π(), and its quotients under various powers of the ideal T of Π which is naturally associated with translations. A central result in the theory is that, in all but one trivial respect, the ring Π/T is actually a graded algebra over . Unfortunately, while the quotients Π/Tk+1 are still graded rings for k > 1, they now only possess a rational algebra structure; to obtain an algebra over , some (weak) continuity assumptions have to be made, although these can be achieved algebraically, by factoring out a further ideal A, the algebra ideal.

A tensor-type integral formula for intrinsic volumes is used to define a further variant of directed projection functions and show that these determine a convex body uniquely. Averages of directed projection functions are then studied, and the connections between the resulting operators and previously considered spherical transforms discussed.

The classical Minkowski sum of convex sets is defined by the sum of the corresponding support functions. The Lp-extension of such a definition makes use of the sum of the pth power of the support functions. An Lp-zonotope Zp is the p-sum of finitely many segments and is isometric to the unit ball of a subspace of ℓq, where 1/p + 1/q = 1. In this paper, a sharp upper estimate is given of the volume of Zp in terms of the volume of Z1, as well as a sharp lower estimate of the volume of the polar of Zp in terms of the same quantity. In particular, for p = 1, the latter result provides a new approach to Reisner's inequality for the Mahler conjecture in the class of zonoids.

In this paper a notion of difference function Δf is introduced for real-valued, non-negative and log-concave functions f defined in Rn. The difference function represents a functional analogue of the difference body K + (−K) of a convex body K. The main result is a sharp inequality which bounds the integral of Δf from above in terms of the integral of f. Equality conditions are characterized. The investigation is extended to an analogous notion of difference function for α-concave functions, with α < 0. In this case also an upper bound for the integral of the α-difference function of f in terms of the integral of f is proved. The bound is sharp in the case α = −∞ and in the one-dimensional case.

Let B (“black”) and W (“white”) be disjoint compact test sets in ℝd, and consider the volume of all its simultaneous shifts keeping B inside and W outside a compact set A ⊂ ℝd. If the union B ∪ W is rescaled by a factor tending to zero, then the rescaled volume converges to a value determined by the surface area measure of A and the support functions of B and W, provided that A is regular enough (e.g., polyconvex). An analogous formula is obtained for the case when the conditions B ⊂ A and W ⊂ AC are replaced by prescribed threshold volumes of B in A and W in AC. Applications in stochastic geometry are discussed. First, the hit distribution function of a random set with an arbitrary compact structuring element B is considered. Its derivative at 0 is expressed in terms of the rose of directions and B. An analogous result holds for the hit-or-miss function. Second, in a design based setting, different random digitizations of a deterministic set A are treated. It is shown how the number of configurations in such a digitization is related to the surface area measure of A as the lattice distance converges to zero.

It is proved that the reciprocal of the volume of the polar bodies, about the Santaló point, of a shadow system of convex bodies Kt, is a convex function of t, thus extending to the non-symmetric case a result of Campi and Gronchi. The case that the reciprocal of the volume is an affine function of t is also investigated and is characterized under certain conditions. These results are applied to prove an exact reverse Santaló inequality for polytopes in ℝd that have at most d + 3 vertices.

This paper treats the L2-discrepancy of digital (0, 1)-sequences over ℤ2, and gives conditions on the generator matrix of such a sequence which guarantee minimal possible order of L2-discrepancy of the generated sequence. The existence is proved for the first time of digital (0; 1)-sequences over ℤ2 with L2-discrepancy of order . This order is best possible by a result of K. Roth. The existence proof is constructive.

By Pick's invariant form of Schwarz's lemma, an analytic function B (z) which is bounded by one in the unit disk D = {z: |z| < 1} satisfies the inequality

at each point α of D. Recently, several authors [2, 10, 11] have obtained more general estimates for higher order derivatives. Best possible estimates are due to Ruscheweyh [12]. Below in §2 we use a Hilbert space method to derive Ruscheweyh's results. The operator method applies equally well to operator-valued functions, and this generalization is outlined in §3.