Mathematical Sciences: Isoperimetric Inequalities

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Erwin Lutwak will continue his investigations of the classical geometric inequalities. Attention will be focussed on establishing a theory dual to the Brunn-Minkowski theory. This latter theory has proved very powerful when applied to problems regarding projections of convex bodies. Lutwak's objective with the dual theory is to develop techniques applicable intersection problems. The Brunn-Minkowski theory of mixed volumes has in recent years been shown to have a dual theory applicable to starshaped sets. This arises essentially from use of the radial function rather than the support function of the set. Many of the classical results for projections and projection bodies have their duals in terms of intersections and intersection bodies. This duality is a useful tool in the study of the apparently anomalous behaviour of intersections.