Cones that are cells, and an application to hyperspaces

Let Y be a compact metric space that is not an (n−1)-sphere. If the cone over Y is an n-cell, then Y×[0,1] is an n-cell; if n≤4, then Y is an (n−1)-cell. Examples are given to show that the converse of the first part is false (for n≥5) and that the second part does not extend beyond n=4. An application concerning when hyperspaces of simple n-ods are cones over unique compacta is given, which answers a question of Charatonik.