A high pressure discharge lamp according to the present invention is a high pressure discharge lamp in which the top ends of a pair of electrodes opposed in a discharge vessel 2 of an arc tube 1 are formed each into a substantially semispherical shape and a protrusion 10 to form an arc spot is formed at each top end, wherein all of the relation: 0.3.cndot..cndot.L/I.cndot..cndot.1.0 for an inter-electrode distance L(mm) between the electrodes and a lamp current I(A) during stable lighting, the relation: 1.2.cndot..cndot.10.sup.-7.cndot..cndot.X/Y.cndot..cndot.1.1.cndot..cndot- .10.sup.-5 for a molar amount X(mol) of bromine in the bromide sealed in the discharge vessel 2 and an inner volume Y(ml) of the discharge vessel 2, and the relation: 9.cndot..cndot.W.sup.3/2/I.cndot..cndot.65, for a weight W(mg) of tungsten at the top end of each electrode 3 formed into a substantially semispherical shape are satisfied, and the lighting frequency of a rectangular wave during stable lighting is 45 Hz or higher.