NOVELTY - A positive electrode active material consists of lithium-nickel-cobalt-metal oxide (1). The median diameter of the positive electrode active material, measured by laser diffraction type particle size distribution measuring apparatus is 2-20 mu m. The average length of primary particle diameter of the particle cross section of the positive electrode active material processed by cross section polisher is 1-10 mu m. The ratio of median diameter and average length of primary particle diameter of the particle cross section of the positive electrode active material is 1-3.    USE - Positive electrode active material is used for lithium secondary battery (claimed) used as power supply for notebook computer, mobile telephone, electric vehicle, and hybrid electric vehicle.    ADVANTAGE - The positive electrode active material provides lithium secondary battery having excellent cycling characteristics.    DETAILED DESCRIPTION - A positive electrode active material consists of lithium-nickel-cobalt-metal oxide of formula: LixNi1-y-zCoyMzO2 (1). The median diameter of the positive electrode active material, measured by laser diffraction type particle size distribution measuring apparatus is 2-20 mu m. The average length of primary particle diameter of the particle cross section of the positive electrode active material processed by cross section polisher is 1-10 mu m, measured by scanning electron microscope (SEM) observation. The ratio of median diameter and average length of primary particle diameter of the particle cross section of the positive electrode active material is 1-3. The diameter of 95% of the positive electrode active material is 3 times or less of median diameter, when the diameter of 5% of cumulative distribution of volume reference are median diameter/3 or more.    M=manganese, aluminum, zirconium, silicon, strontium, or magnesium;    x=0.95-1.10;    y=0.1-0.4;and    z=0-0.1.    An INDEPENDENT CLAIM is included for manufacture of the positive electrode active material, which involves mixing lithium hydroxide with nickel-hydroxide particles containing cobalt element, primarily baking the obtained mixture at 730-950 degrees C in oxidative atmosphere, washing the baked product using water, and removing excessive lithium from the baked product, and secondarily baking the baked product at 600-900 degrees C. The molar ratio of lithium with respect to sum total of cobalt and nickel is 1.5-5.