  We study new classes of convex bodies and star bodies with unusual
properties. First we define the class of reciprocal bodies, which may be viewed
as convex bodies of the form "$1/K$". The map $K\mapsto K^\prime$ sending a
body to its reciprocal is a duality on the class of reciprocal bodies, and we
study its properties.
  To connect this new map with the classic polarity we use another
construction, associating to each convex body $K$ a star body which we call its
flower and denote by $K^\clubsuit$. The mapping $K\mapsto K^\clubsuit$ is a
bijection between the class $\mathcal{K}_0^n$ of convex bodies and the class
$\mathcal{F}^n$ of flowers. We show that the polarity map
$\circ:\mathcal{K}_0^n\to\mathcal{K}_0^n$ decomposes into two separate
bijections: First our flower map $\clubsuit:\mathcal{K}_0^n\to\mathcal{F}^n$,
followed by the spherical inversion $\Phi$ which maps $\mathcal{F}^n$ back to
$\mathcal{K}_0^n$. Each of these maps has its own properties, which combine to
create the various properties of the polarity map.
  We study the various relations between the four maps $\prime$, $\circ$,
$\clubsuit$ and $\Phi$ and use these relations to derive some of their
properties. For example, we show that a convex body $K$ is a reciprocal body if
and only if its flower $K^\clubsuit$ is convex.
  We show that the class $\mathcal{F}^n$ has a very rich structure, and is
closed under many operations, including the Minkowski addition. This structure
has corollaries for the other maps which we study. For example, we show that if
$K$ and $T$ are reciprocal bodies so is their "harmonic sum"
$(K^\circ+T^\circ)^\circ$. We also show that the volume
$\left|\left(\sum_i\lambda_{i}K_i\right)^\clubsuit\right|$ is a homogeneous
polynomial in the $\lambda_i$'s, whose coefficients can be called
"$\clubsuit$-type mixed volumes". Related geometric inequalities are also
derived.
