Symmetric cone

Remarks

Example: the positive orthant $\Omega = (0, \infty)^n$ in $\R^n$ with the dot product. A vector $y$ satisfies $\ip{x}{y} > 0$ for every nonzero $x$ with nonnegative entries exactly when every entry of $y$ is positive, so $\Omega^* = \Omega$. The diagonal matrix with entries $y_i / x_i > 0$ maps $x \in \Omega$ to $y \in \Omega$ and preserves $\Omega$, so $\Omega$ is homogeneous.

Non-example: the sector $\Omega = \set{(s, t) \in \R^2 : \abs{t} < 2s}$ with the dot product. It is homogeneous, since a linear map that scales its two boundary rays by independent positive factors preserves it and moves any interior point to any other. It is not self-dual: its opening angle is $2 \arctan 2 > \pi/2$, while its dual cone has the opening angle $\pi - 2\arctan 2 < \pi/2$.

Other standard examples are the Lorentz cone $\set{(t, x) \in \R \times \R^{n} : t > \norm{x}}$ and the cone of positive definite real symmetric $n \times n$ matrices with the inner product $\ip{A}{B} = \tr(AB)$.