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Convexity and quasiconvexity are two properties that capture the concept ofdiversification for risk measures. Between the two, there is naturalquasiconvexity, an old but not so well-known property weaker than convexity butstronger than quasiconvexity. A detailed discussion on natural quasiconvexityis still missing and this paper aims to fill this gap in the setting ofconditional risk measures. We relate natural quasiconvexity to additivelydecomposable sums. The notion of convexity index, defined in 1980s forfinite-dimensional vector spaces, plays a crucial role in the discussion ofdecomposable sums. We propose a general treatment of convexity index intopological vector spaces and use it to study naturally quasiconvex riskmeasures. We prove that natural quasiconvexity and convexity are equivalent forconditional risk measures on $L^p$ spaces, $p \geq 1$, under mild continuityand locality conditions. Finally, we discuss an alternative notion of localitywith respect to an orthonormal basis in $L^2$.

Decomposable sums and their implications on naturally quasiconvex risk measures