Abstract
The family of convex sets in a (finite dimensional) real vector space admits several unary and binary operations – dilatation, intersection, convex hull, vector sum – which preserve convexity. These generalize to convex functions, where there are in fact further operations of this kind. Some of the latter may be regarded as combinations of two such operations, acting on complementary subspaces. In this paper, a general theory of such mixed operations is introduced, and some of its consequences developed.
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References
Billera, L. J. and Sturmfels, B.: Fiber polytopes, Ann. of Math. 135 (1992), 527-549.
Goodey, P. R., Kiderlen, M. and Weil, W.: Section and projection means of convex bodies, Monatsh. Math. 126 (1998), 37-54.
Goodey, P. R. and Weil, W.: The determination of convex bodies from the mean of random sections, Math. Proc. Cambridge Philos. Soc. 112 (1992), 419-430.
Grünbaum, B.: Convex Polytopes, Wiley-Interscience, New York, 1967.
Rockafellar, R. T.: Convex Analysis, Princeton Univ. Press, 1970.
Schneider, R.: Convex Bodies: the Brunn-Minkowski Theory, Cambridge Univ. Press, 1993.
Ziegler, G. M.: Lectures on Polytopes, Graduate Texts in Mathematics 152, Springer-Verlag, 1995.
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McMullen, P. New Combinations of Convex Sets. Geometriae Dedicata 78, 1–19 (1999). https://doi.org/10.1023/A:1005019802841
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DOI: https://doi.org/10.1023/A:1005019802841