C. M. Petty has conjectured the minimum value for a certain affine‐invariant functional denned on the class of convex bodies. We give sharp bounds for this functional on a certain subclass of convex bodies, and we give a counterexample to an upper bound proposed by R. Schneider for the class of centrally symmetric convex bodies. We conjecture that the simplex provides the maximum on the class of all convex bodies, while the largest centrally symmetric subset of a simplex gives a sharp upper bound on the class of all centrally symmetric convex bodies.

Abstract. Two strengthenings of the Wills conjecture, an extension of Bonnesen's inradius inequality to n-dimensional space, are obtained. One is the sharpest of the known strengthenings of the conjecture in three dimensions; the other employs techniques which are fundamentally different from those utilized in the other proofs.One of the best known geometric inequalities is the isoperimetric inequality, which states that of all closed planar curves with fixed perimeter, a circle encloses the greatest area. If K is a planar convex body (a convex "body" is a compact convex set with nonempty interior) of perimeter L with area A, then this inequality can be expressed aswith equality only when the boundary of K is a circle.A strengthening of (1) was given by the Danish mathematician T. Bonnesen, who demonstrated in 1929 (see [5]) that if K has circumradius R and inradius r, thenThis inequality can be derived from the inequalityThe inequality (3) with λ = r is known as Bonnesen's inradius inequality.J. M. Wills [15] conjectured in 1970 thatwhich would be an extension of Bonnesen's inradius inequality to higher dimensions.

The projection body is determined for selected three-dimensional convex bodies. The relationship between the volume of a convex body and the volume of its projection body is explored by calculating the value of an affine-invariant functional defined on the class of convex bodies, and conjectures are made as to the significance of these calculations.

No abstract

No abstract