“They have always called me crazy,” he said. “The thing is, crazy are the people who don’t understand that the future belongs to those who are not standard, who are open and vulnerable.”

Still, even in a relatively tame, 45-minute discussion in his office in Kiev, Mr. Chernovetsky seemed a little off. For much of the interview, he would not look at his questioner, speaking in almost a monotone while staring at the floor.

In public appearances, this bearing  combined with occasional bursts of exuberance  has prompted speculation that he abuses alcohol or drugs, or is on heavy medication. At one event promoting religious diversity, he slurred his language, pausing at length as he fumbled for the words for Judaism and Hinduism.

HE has denied the rumors about drugs and alcohol use, though it has not helped that he has taken short, unexplained leaves of absence.

Mr. Chernovetsky, who is married with two children, was elected in 2006 after supporting the Orange Revolution, and became known for courting retired people by strolling neighborhoods and handing out flour and other food.

There are three major political factions in Ukraine, led by President Viktor A. Yushchenko, Prime Minister Yulia V. Tymoshenko and Viktor F. Yanukovich, a former prime minister. Mr. Chernovetsky, an independent, has at times built alliances with each. All three factions covet the mayor’s office but fear that removing Mr. Chernovetsky through something akin to impeachment would allow a rival’s candidate to take over.

In 2008, the Parliament, led by Ms. Tymoshenko, tried to unseat Mr. Chernovetsky by calling a special election. He won, after his opponents could not unify around a candidate and split the vote. He has since suggested that he might run for president.