Three cuts, seven equal pieces

By extending the edges of a triangle, one can cut a cake into seven pieces. Is there a convex shape that can be cut into 7 equal-area pieces by 3 cuts? Failing that, of all shapes which can be cut into 7 equal-area pieces by 3 cuts, which has the minimum perimeter? Does the answer change if we prohibit holes? The answer will probably have a pleasing 3-pointed boomerang shape. Is it possible to cut a (possibly concave) shape into 7 equal-area pieces by 3 cuts so that the resulting pieces are all convex? (Pretty easily yes.) What if we require every cut to be of the form "air-cake-air"? That is, the knife encounters only a single segment of cake along the infinite length cut.