Illumination bodies for ball-convex bodies

We introduce illumination bodies and weighted illumination bodies in the class of $n$-dimensional $R$-ball convex bodies. These bodies may be regarded as dual counterparts of the recently introduced $R$-ball floating bodies. We prove that the illumination bodies are convex. We also show that a left derivative of volume gives rise to a surface area measure for ball-convex bodies, the relative surface area measure. In dimension $2$, we establish an isoperimetric inequality for this relative surface area.