* Lucie Arnaz, her daughter with Desi Arnaz, is a successful actress, singer, and producer. She's also a strong advocate for her mother's legacy and has been involved in numerous projects related to Lucille Ball's life and career.
* Desi Arnaz Jr., her son with Desi Arnaz, had a successful career in the 1960s and 70s, primarily as a musician and actor. He also appeared in the film "Lucy, Daughter of the Sun" with his mother.
* **Some of Lucille Ball's grandchildren have pursued careers in the entertainment industry, but not on the same scale as their parents. For example, her granddaughter, Katey Sagal, is a well-known actress, but she's not primarily associated with Lucille Ball's legacy.
It's important to note that "showbiz" is a broad term, and some of Lucille Ball's grandchildren may have had some involvement in the industry, even if they didn't achieve mainstream fame.
To get a complete picture, it would be helpful to know which specific grandchildren you're interested in.