The incident happened just after 4 p.m. Tuesday at the store located on the corner of La Cienega Boulevard and Centinela Avenue.
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According to LAPD Capt. Kelly Muniz, an 83-year-old man shot a 31-year-old man during a dispute involving a woman.
The verbal argument escalated when the 83-year-old took out a gun and shot the victim.
The 31-year-old, whose identity has not been released, was pronounced dead at the scene.
The suspect has since been taken into custody. There's no word on if the woman involved was injured.
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Witnesses at the scene told Eyewitness News, the suspect, a Korean War veteran, shot the 31-year-old who was reportedly arguing with the woman after "racial slurs" were said.
"We thought it was over because they led him away," witness Tracey Knox said. "They turned him around. 'Go back into your house,' it looked like they were saying to him."
She said the younger man took off his shirt and said, "I'm from South Central, (expletive)!"
"And that old man just turned around -- bam. One shot," Knox said. "Impeccable. One shot."
The incident remains under investigation.