Final answer:
Coco Chanel is the designer known for designing the Little Black Dress and having Audrey Hepburn as the luxury brand's muse.
Step-by-step explanation:
The designer known for designing the Little Black Dress and having actress Audrey Hepburn as the luxury brand's muse for over 40 years is Coco Chanel. Chanel was a prominent French fashion designer who made significant contributions to the world of fashion during the early 20th century. Her iconic designs and innovative approach to women's clothing revolutionized the industry and continue to be influential today.