In this paper, we present an explicit geometric characterization of quadratically stabilizing state feedback laws that are based on the use of multivariable quantizers of minimum dimension. This characterization consists of a set of necessary and sufficient conditions for a quantized static state feedback to render a given quadratic function a Lyapunov function for the closed-loop system. These necessary and sufficient conditions provide a means to analyse and design such quantized feedback laws and are derived from set inclusion conditions that are necessarily satisfied by the quantization regions and values of a quadratically stabilizing quantizer.