Inequalities for mixed projection bodies

Abstract

Abstract. Mixed projection bodies are related to ordinary projection bodies (zonoids) in the same way that mixed volumes are related to ordinary volume. Analogs of the classical inequalities from the Brunn-Minkowski Theory (such as the Minkowski, Brunn-Minkowski, and Aleksandrov-Fenchel inequalities) are developed for projection and mixed projection bodies.Two decades ago, Bolker [5] observed that projection bodies (also known as zonoids) were objects of independent investigation in a number of mathematical disciplines such as measure theory, crystallography, optimal control theory, functional analysis, and geometric convexity. Since the appearance of Bolker's article, projection bodies have received considerable increased attention (see, for example, [7,13,14,21,23,26,27,28,32,40, 41] Mixed projection bodies are related to ordinary projection bodies in the same way that mixed volumes are related to ordinary volume. The definition and some elementary properties of mixed projection bodies can be found in the classic volume of . Support functions of mixed projection bodies were investigated by Chakerian [9]. Stability questions for mixed projection bodies are treated by Goodey [11] and . In [18] and [19], inequalities for the polars of mixed projection bodies were obtained. This article treats the corresponding inequalities for the mixed projection bodies themselves. Analogs of the classical mixed volume inequalities (such as the Brunn-Minkowski, Minkowski, and Aleksandrov-Fenchel inequalities) will be established for mixed projection bodies.Since interest in zonoids is not limited to one discipline, an attempt is made to make this article reasonably self-contained.Background material and notation regarding mixed volumes and mixed surface area measures is given in §0. The classical mixed volume inequalities are

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