Italy has the Agnellis; France has the Arnaults, and patriarch Bernard Arnault is usually in the top three—now and again in pole position—on the league table of the world’s richest men. As well as running an empire that includes Christian Dior, Louis Vuitton, and a range of brands from Clos des Lambrays, a vineyard founded in the 14th century, to Rihanna’s Fenty, he also plays the piano and enjoys knocking a tennis ball about with Roger Federer.

The Arnault family lives up to the image of the products on which Bernard Arnault built his fortune. Delphine Arnault is the poised executive V.P. of Louis Vuitton. Smolderingly good-looking Antoine Arnault, married to model and philanthropist Natalia Vodianova, helms Berluti and Loro Piana. Alexandre bought and runs Rimowa. And, in June, yet another Arnault landed a high-profile job in the family business when 25-year-old Frédéric Arnault was appointed C.E.O. of Tag Heuer.

Frédéric, back left, with his father, Bernard Arnault, and brother Jean, front left, at the men’s singles final of the 2016 French Open, at Roland-Garros.

Frédéric has a girlfriend, but he does a good job at keeping her identity under wraps. He does his bit turning up at galas, vernissages, and so on, but paparazzi pictures of him betray that faint awkwardness of a clever person being photographed doing something trivial. And he is clever: after finishing his business studies in Paris, and following a research internship at Facebook, he considered pursuing a Ph.D. in artificial intelligence.

It can be safely said that he favors a cerebral environment. He is much more comfortable at the Pianoscope festival in Beauvais—not in the audience but onstage. Seeing him at the piano is to watch him enter another world. Last year at Beauvais, he played a Liszt étude, Chopin’s Ballade No. 1, and the Liszt Sonata in B Minor. “It is very difficult, but it is my favorite piece of music,” he says. “It is deep, complex, and, like a theater play, it was inspired by Goethe’s Faust.” He plays golf too, but he is careful not to play it overly well. “We have one rule in the group,” he explains. “If your handicap is below 10, it means you don’t work enough.”

He does his bit turning up at galas, vernissages, and so on, but paparazzi pictures of him betray that faint awkwardness of a clever person being photographed doing something trivial.

Frédéric recently launched a golf-specific smartwatch for Tag Heuer, but is also careful to stress his commitment to the historic business of mechanical watches. Still, he does not give too much away about his plans, taking refuge in generalizations such as “investing in quality, savoir faire, and disruptive communications.” He is similarly opaque on the LVMH takeover of Tiffany & Co. “I’m really 100 percent focused on Tag Heuer,” he demurs. And he also avoids discussing succession planning at LVMH. “I’m not the right person to answer this question,” he says flatly, before adding, “There’s no written plan.”

Back to the playing field. Of all of his family members, I wonder, which is the most ferocious opponent for Federer? “I have to admit that it’s Jean,” he says, referring to his youngest brother, who has just finished studying mechanical engineering at Imperial College London, and will enroll at M.I.T. this autumn. “It’s often the case in families,” he says. “It’s the youngest who’s the best at sport.”

And is that the case when it comes to business too? He pauses. “In business, there are fewer examples.”

Nicholas Foulkes is a writer based in London