Collections of functions forming a partition of unity play an important role in analysis. In this paper we characterise for any $N\in \mathbb{N}$ the entire functions $P$ for which the partition of unity condition $\sum _{\mathbf{n}\in \mathbb{Z}^{d}}P(\mathbf{x}+\mathbf{n})\unicode[STIX]{x1D712}_{[0,N]^{d}}(\mathbf{x}+\mathbf{n})=1$ holds for all $\mathbf{x}\in \mathbb{R}^{d}.$ The general characterisation leads to various easy ways of constructing such entire functions as well. We demonstrate the flexibility of the approach by showing that additional properties like continuity or differentiability of the functions $(P\unicode[STIX]{x1D712}_{[0,N]^{d}})(\cdot +\mathbf{n})$ can be controlled. In particular, this leads to easy ways of constructing entire functions $P$ such that the functions in the partition of unity belong to the Feichtinger algebra.

A geometric property of convex sets which is equivalent to a minimax inequality of the Ky Fan type is formulated. This property is used directly to prove minimax inequalities of the von Neumann type, minimax inequalities of the Fan-Kneser type, and fixed point theorems for inward and outward maps.