In this article we present topics that can be considered when developing a landscape of investigation that reflects on the issues of the LGBTQ+ community. We outline the landscape "LGBTQ+ Life Conditions," having prospective teachers in mind. The outlining takes place in four steps: (1) We indicate the relevance of this topic and bring in real-life complexities into the landscape of investigation to engage the students for doing investigations with mathematics connected to this topic; (2) we present some statistical data that show the relevance of researching the topic and the potentials of mathematics for doing so; (3) we outline some possible investigations when exploring the landscape, taking inspiration from already carried out investigations as well as from some imagined investigations; (4) we reflect on some challenges and possibilities for those who want to explore this landscape in mathematics classes.