Let K be a convex body in R and δ > 0. The homothety conjecture asks: Does Kδ = cK imply that K is an ellipsoid? Here Kδ is the (convex) floating body and c is a constant depending on δ only. In this paper we prove that the homothety conjecture holds true in the class of the convex bodies B p , 1 ≤ p ≤ ∞, the unit balls of l p ; namely, we show that (B p )δ = cB p if and only if p = 2. We also ...

A convex body K in R has associated with it a unique circumscribed ellipsoid CE(K) with minimum volume, and a unique inscribed ellipsoid IE(K) with maximum volume. We first give a unified, modern exposition of the basic theory of these extremal ellipsoids using the semi–infinite programming approach pioneered by Fritz John in his seminal 1948 paper. We then investigate the automorphism groups o...

We establish the greatest upper bound for the product of the i-th brightness-integrals of a convex body and its polar dual. Further, the greatest upper bound for the product of the brightness-integrals of order p of a convex body and its polar dual is also given.

We propose strongly consistent algorithms for reconstructing the characteristic function 1K of an unknown convex body K in R from possibly noisy measurements of the modulus of its Fourier transform 1̂K . This represents a complete theoretical solution to the Phase Retrieval Problem for characteristic functions of convex bodies. The approach is via the closely related problem of reconstructing K ...

The volumetric method of 'convex hulling' has recently been put forward as a mass prediction technique for fossil vertebrates. Convex hulling involves the calculation of minimum convex hull volumes (vol(CH)) from the complete mounted skeletons of modern museum specimens, which are subsequently regressed against body mass (Mb) to derive predictive equations for extinct species. The convex hullin...

Let Π be the projection operator, which maps every polytope to its projection body. It is well known that Π maps the set of polytopes, P, in R into P, that it is a valuation, and that for every P ∈ P, ΠP is affinely associated to P . It is shown that these properties characterize the projection operator Π. This proves a conjecture by Lutwak. Let Kn denote the set of convex bodies (i.e., of comp...