So there was bound to be trouble last week when the Italian environmentalist group Legam-biente had the temerity to bring the worst kind of news that an elegant resort like Capri could ever hear.
The island's beaches, the group said, were dirty. Not just dirty, in fact, but in some cases disgusting. At one spot, just a stone's throw from the yacht belonging to the Greek shipping magnate Stavros Niarchos, the sea was apparently teeming with more than 10 times the accepted level of coliform bacteria.
In layman's terms, that meant someone was pumping raw sewage straight out of the island's drains into the Mediterranean.
Well, there was only one response to that kind of calumny, and the mayor, Costantino Federico, wasted no time in delivering it. "We don't need to be saved by anyone, least of all a group of ecologists," he fumed. "Our waters are blue, as the samples taken by the health authorities show. Those of us who love this island can sleep easy."
In no time, a full-scale row was brewing. Legambiente challenged the mayor to issue an official declaration that the sea was clean. The mayor said the group's figures were simply wrong and he was not prepared to discuss them further.
Then a local ecologist revealed that a sewage tank at Marina Piccola, on the south side of the island, had broken in May and had not been replaced, causing its contents to spill into the sea. Furthermore, he said, the island did not have a proper waste treatment system.
So the argument went on. Was this another piece of bad news for the Bay of Naples tourist industry in a week already marked by rumours of an imminent eruption of Mount Vesuvius? Would British holidaymakers - already scared away from Sorrento and the Amalfi coast by the claim of a Neapolitan vulcanologist that the region was a "disaster waiting to happen" and predicting a big eruption within 10 years - back out of Capri too?
In all the furore, nobody pointed out the hard truth: that Capri has been polluted for years. Sitting on the southern tip of the Bay of Naples, it is near some of the dirtiest water in the Mediterranean. Granted, the island is protected to some extent by favourable currents, but on a bad day the sea is awash with plastic bottles, sodden cigarette butts and who knows what chemical residues from the Naples metropolitan area.
Of course, the rich and famous in Capri's flower-filled villas know this perfectly well. You would never catch the likes of Ivana Trump or Walid Jumblatt diving into the sea off the island's beautiful rocky coastline. They keep to the seclusion of their swimming-pools, or sail their luxurious yachts into the wide blue yonder.
In fact, the days when Capri was a rich man's retreat are long gone. More than a million tourists mob the place each year, spending little at the island's overpriced bars and restaurants but still liberally dumping their litter before taking the return ferry back to Naples. Most of the jet set ran away years ago.
The myth of the island's exclusivity is nevertheless fondly nurtured, particularly by the mayor, who has devoted much energy to raising the tone. In the late 1980s he issued a flurry of ordinances, banning naked torsos and sleeping-bags and declaring certain spots off-limits to tour groups.
More recently, Mr Federico has suggested declaring Capri as an independent state, maybe a monarchy. He even proposed Princess Stephanie of Monaco for the job of Queen of Paradise.
To some extent his efforts have paid off, and celebrities have been dribbling back. Last week, for example, the King and Queen of Belgium turned up. No wonder the mayor is so sensitive to criticism: after all, distasteful talk of raw sewage is not the thing to entice a better class of tourist.