Minkowski sum of convex sets in the plane which are not polygons

Can the Minkowski sum of two convex sets in the plane which are not polygons be a polygon? Explicitly my convex set is of the form $C=\{(x,y)\in {\mathbb{R}}^{2}:x,y\ge 0\text{,}\sqrt{y}+\sqrt{x}\ge 1\text{and}y\le 1-x\}$

I am interested in knowing if there is a convex set C' such that the Minkowski sum

$C+{C}^{\prime}=P$

where $P=\{(x,y)\in {\mathbb{R}}^{2}:x,y\ge 0\text{and}1/2\le x+y\le 1\}.$

Can the Minkowski sum of two convex sets in the plane which are not polygons be a polygon? Explicitly my convex set is of the form $C=\{(x,y)\in {\mathbb{R}}^{2}:x,y\ge 0\text{,}\sqrt{y}+\sqrt{x}\ge 1\text{and}y\le 1-x\}$

I am interested in knowing if there is a convex set C' such that the Minkowski sum

$C+{C}^{\prime}=P$

where $P=\{(x,y)\in {\mathbb{R}}^{2}:x,y\ge 0\text{and}1/2\le x+y\le 1\}.$