Which sin is worse if one had no choice but to commit one of the two: a negiah violation or a Chillul Hashem? In my business life, I routinely cross paths with women who expect a handshake or sometimes a hug (a non-romantic one that is just a social gesture to which they are accustomed). Failing to do so is something they consider highly offensive in their culture. Many of these women I interact with know my language very sparsely, making explaining my religious beliefs to them nearly impossible.