Proposition 2 — $\rho$ is star-shaped iff its acceptance set is star-shaped

Let $\rho:\mathcal X\to\mathbb R$ be a risk measure on a space $\mathcal X$ of bounded positions containing the constants. Recall that a subset $S$ of a vector space is star-shaped if $\lambda s\in S$ for all $\lambda\in[0,1]$ and all $s\in S$. The following are equivalent:

1. $\rho$ is star-shaped;
2. the acceptance set $\mathcal A_\rho=\{X\mid\rho(X)\le0\}$ is star-shaped in $\mathcal X$;
3. there exists a star-shaped acceptance set $\mathcal A$ with $\rho=\rho_{\mathcal A}$, where $\rho_{\mathcal A}(X)=\inf\{m\mid X-m\in\mathcal A\}$.

This is the principle behind star-shapedness: deleveraging an acceptable position keeps it acceptable.

Formalization Note Star-shaped sets are Mathlib's StarConvex ℝ 0, which is exactly the page's definition. Acceptance sets carry both axioms (least upper bound $0$ of the constants, downward closedness).