On projection bodies of order one

Abstract. The projection body of order one of a convex body K in R^n is the body whose support function is, up to a constant, the average mean width of the orthogonal projections of K onto hyperplanes through the origin. The paper contains an inequality for the support function of the projection body of order one of K which implies in particular that such a function is strictly convex, unless K has dimension one or two. Furthermore, an existence problem related to the reconstruction of a convex body is discussed to highlight the different behavior of the area measures of order one and of order n − 1.