The ambient space if ${\bf M}_n({\mathbb R})$.

Let us begin with facts. 1- The cone of positive semi-definite symmetric matrices is convex. 2- It is a little subtler that the cone $K^+$ of matrices with non-negative determinant is rank-one convex; this is because $\det(M+xy^T)=\det M+x^T\hat My$, where $\hat M$ is the cofactor matrix. Therefore, whenever $\det A,\det B\ge0$ and $B-A$ is rank-one, then the determinant is $\ge0$ over the segment $(A,B)$.

Is there an interesting example (say, a non-convex one) of a "singularly convex" cone ? By singularly convex, I mean that if $A,B\in K$ and $\det(B-A)=0$, then $(A,B)\subset K$.

Of course, if $n=2$, singular convexity is just rank-one convexity.

Mind that I am specially interested in cones formed of symmetric matrices.