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We study a combinatorial notion where given a set S of lattice points one takes the set of all sums of p distinct points in S, and we ask the question: ‘if S is the set of lattice points of a convex lattice polytope, is the resulting set also the set of lattice points of a convex lattice polytope?’ We obtain a positive result in dimension 2 and a negative result in higher dimensions. We apply this to the corner cut polyhedron.
In this paper, we explore affine semigroup versions of the convex geometry theorems of Helly, Tverberg, and Carathéodory. Additionally, we develop a new theory of colored affine semigroups, where the semigroup generators each receive a color and the elements of the semigroup take into account the colors used (the classical theory of affine semigroups coincides with the case in which all generators have the same color). We prove an analog of Tverberg’s theorem and colorful Helly’s theorem for semigroups, as well as a version of colorful Carathéodory’s theorem for cones. We also demonstrate that colored numerical semigroups are particularly rich by introducing a colored version of the Frobenius number.
covering with convex sets, Handbook of convex geometry , Vol. A, B . North-Holland, Amsterdam , 1993 , pp. 799 – 860 . [6] L . Fejes Tóth . Research Problems: Exploring a Planet . Amer. Math. Monthly , 80 ( 9 ): 1043 – 1044 , 1973 . [7] F . Fodor
Introduction to Convex Geometry with Applications . Birkhäuser / Springer, Cham , 2019 . [18] V. V . Prasolov . Problems in Plane Geometry , 5 th ed., rev. and compl. (Russian) . Moscow, Russia : The Moscow Center for Continuous Mathematical Education
. Groemer . On the Brunn–Minkowski theorem . Geom. Dedicata , 27 : 357 – 371 , 1988 . [23] H . Groemer . Stability of geometric inequalities. In: Handbook of convex geometry ( P. M . Gruber , J. M . Wills , eds), North-Holland, Amsterdam , 1993
-1-2011-0471 and CONACYT 121158 “Discrete and Convex Geometry”, and by Hungarian OTKA grant No. 75016. This paper was supported by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences. Communicated by G. Fejes
