abstract: Guy Bouchitté formulated the "Vanishing Mass Conjecture" about twenty years ago in connection to a problem of shape optimization of light structures; to my knowledge the only progress in this direction has been obtained by Babadjian, Iurlano and Rindler only this year. In this talk I will illustrate this conjecture placing the emphasis on the underlying geometric nature, and explore the connections to other results and questions of geometric (and measure-theoretic) flavor.