Under the condition of maintaining the network connectivity, each sensor varies adaptively its transmission power, which can minimize the power consumption and reduce the radio interference, so as to prolong the lifetime of wireless sensor networks. A new geometry structure briefly named PSLDel is presented based on the Voronoi tessellation and the local Delaunay triangulation, and the efficient distributed algorithms to construct the new structure are proposed based on the message exchange. With PSLDel as the underlying logical topology of wireless sensor networks, each sensor can adjust its transmission power to minimum according to the farthest logical neighbor, which forms a network topology with desirable features, such as connectivity, sparseness, planar and t-spanner. Theoretic analyses and simulation results show that the performance of PSLDel is close to that of UDel, which is globally constructed, in terms of the logical neighbor, the minimal transmission power and the radio interference, but the network delay of PSLDel is slightly better than that of UDel. Meanwhile, compared with AUDel, which is another approximate structure of UDel and can also be locally constructed, the communication cost of constructing PSLDel can be reduced by 55 percent at least. Thus the properties can improve the energy efficiency and prolong the lifetime of wireless sensor networks.