Commercial and residential buildings account for a large share of any country's energy consumption (e.g., 40% of total energy consumption in the United States). Within buildings, a major portion of the expended energy is used to provide heating and cooling. However, most buildings are heated and cooled regardless of the occupants' locations, presenting a significant opportunity for improving their energy efficiency. Zonal heating and cooling systems eliminate some of this waste, but are still limited by the large granularity of zones (e.g., floors of a building). In this paper, we propose a localized heating system that significantly reduces energy consumption due to heating. Our system utilizes radiant heaters to provide directed heat to occupants and enables the temperature setting in the central heating system to be lowered, resulting in significant net energy savings. To appropriately direct radiant heating, our system utilizes a minimally-intrusive indoor positioning system capable of locating the occupants. We analyze the heating energy and cost savings achievable from the proposed system using a representative set of building models in the EnergyPlus simulation framework. Our analysis indicates that the proposed localized heating system can provide average savings of 52% in heating energy (maximum of 76%) and 46% in heating cost (maximum of 59%) when compared to a system that uses central heating alone. Our results highlight the potential of localized heating techniques in enhancing building energy efficiency.