Discrete Applied Mathematics 160 (2012) 588–592
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Discrete Applied Mathematics
journal homepage: www.elsevier.com/locate/dam
Sperner’s lemma and zero point theorems on a discrete simplex and a
, Kazuo Murota
, Akihisa Tamura
School of Business Administration, Tokyo Metropolitan University, Tokyo 192-0397, Japan
Department of Mathematical Informatics, Graduate School of Information Science and Technology, University of Tokyo, Tokyo 113-8656, Japan
Department of Mathematics, Keio University, Yokohama 223-8522, Japan
a r t i c l e i n f o
Received 25 October 2010
Received in revised form 9 November 2011
Accepted 17 November 2011
Available online 7 December 2011
a b s t r a c t
We show two discrete zero point theorems that are derived from Sperner’s lemma and
a Sperner-like theorem (van der Laan and Talman [Math. Oper. Res. (1982)] ; Freund
[Math. Oper. Res. (1986)]) . Applications to economic and game models are also pre-
© 2011 Elsevier B.V. All rights reserved.
In this paper, we show two zero point theorems for a certain (meaningful) class of correspondences, one on the discrete
simplex and the other on the discrete simplotope (see Section 2 for the precise definitions). The two theorems are closely
related to the discrete fixed point theorem in  and its generalization [7,1,6], which all exploit some sort of ‘‘direction
preserving’’ condition. In this paper, we will use a new, weaker, ‘‘simplexwise’’ version of it, restricting the domains and
images of correspondences. The two discrete zero point theorems are, loosely speaking, the contrapositions of Sperner’s
lemma on the discrete simplex and of a Sperner-like theorem [5,3] on the discrete simplotope, respectively. As such,
they will also provide us with a combinatorial viewpoint for the existing types of discrete fixed or zero point theorems
[4,7,1,6]. We also believe that our class of correspondences is useful for applications, as a means to represent a field of price
adjustment, strategy adjustment, or something that has a flavor of vector fields.
In Section 2, we give some definitions and lemmas. Section 3 gives the theorems and Section 4 gives applications to
economic and game models.
2. Definitions and lemmas
be the d-dimensional Euclidean space and Z
the set of its integer points. We denote by 0 a zero vector, x · y the
inner product of vectors x, y ∈ R
, and conv X the convex hull of a set X ⊆ R
. A (d − 1)-dimensional simplex is a set
This work is partially supported by the Grants-in-Aid for Scientific Research (B) and (C) from JSPS, by the Global COE ‘‘The Research and Training Center
for New Development in Mathematics’’, and by the Aihara Project, the FIRST program from JSPS, initiated by CSTP. We thank Jun Wako and anonymous
reviewers for their valuable comments.
Corresponding author. Fax: +81 42 677 2298.
E-mail addresses: email@example.com (T. Iimura), firstname.lastname@example.org (K. Murota), email@example.com (A. Tamura).
0166-218X/$ – see front matter © 2011 Elsevier B.V. All rights reserved.