Abstract
We show that ifP⊂\(\mathbb{E}^d \), |P|=d+k,d⩾k⩾1 andO ∈ int convP, then there exists a simplexS of dimension ⩾\(\left[ {\frac{d}{k}} \right]\) with vertices inP, satisfyingO ∈ rel intS, the bound being sharp. We give an upper bound for the minimal number of vertices of facets of a (j-1)-neighbourly convex polytope in\(\mathbb{E}^d \) withv vertices.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bezdek, A., Bezdek, K., Makai, E. Jr., McMullen, P.: Facets with fewest vertices. Mh. Math.109, 89–96 (1990).
Gale, D.: Neighbouring vertices on a convex polyhedron, linear inequalities and related systems. In: Linear Inequalities and Related Systems, pp. 255–263. (H. W. Kuhn, A. W. Tucker, eds.) Princeton: Univ. Press. 1956.
McMullen, P., Shephard, G. C.: Convex Polytopes and the Upper Bound Conjecture. London-New York: Cambridge Univ. Press. 1971.
Steinitz, E.: Bedingt konvergente Reihen und konvexe Systeme II. J. Reine Angew. Math.144, 1–40 (1914).
Author information
Authors and Affiliations
Additional information
Research (partially) supported by Hung. Nat. Found. for Sci. Research, grant no. 1817
Research (partially) supported by Hung. Nat. Found. for Sci. Research, grant no. 326-0213
Rights and permissions
About this article
Cite this article
Bezdek, A., Bezdek, K. & Makai, E. Interior points of the convex hull of few points in\(\mathbb{E}^d \) . Monatshefte für Mathematik 111, 181–186 (1991). https://doi.org/10.1007/BF01294265
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01294265