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General definition and special cases

Let X be a locally convex Hausdorff vector space, and consider a probability measure μ on the Borel σ-algebra of X. Fix −∞ ≤ s ≤ 0, and define, for u, v ≥ 0 and 0 ≤ λ ≤ 1,

M s , λ ( u , v ) = { ( λ u s + ( 1 − λ ) v s ) 1 / s if  − ∞ < s < 0 , min ( u , v ) if  s = − ∞ , u λ v 1 − λ if  s = 0. {\displaystyle M_{s,\lambda }(u,v)={\begin{cases}(\lambda u^{s}+(1-\lambda )v^{s})^{1/s}&{\text{if }}-\infty <s<0,\\\min(u,v)&{\text{if }}s=-\infty ,\\u^{\lambda }v^{1-\lambda }&{\text{if }}s=0.\end{cases}}}

For subsets A and B of X, we write

λ A + ( 1 − λ ) B = { λ x + ( 1 − λ ) y ∣ x ∈ A , y ∈ B } {\displaystyle \lambda A+(1-\lambda )B=\{\lambda x+(1-\lambda )y\mid x\in A,y\in B\}}

for their Minkowski sum. With this notation, the measure μ is said to be s-convex³ if, for all Borel-measurable subsets A and B of X and all 0 ≤ λ ≤ 1,

μ ( λ A + ( 1 − λ ) B ) ≥ M s , λ ( μ ( A ) , μ ( B ) ) . {\displaystyle \mu (\lambda A+(1-\lambda )B)\geq M_{s,\lambda }(\mu (A),\mu (B)).}

The special case s = 0 is the inequality

μ ( λ A + ( 1 − λ ) B ) ≥ μ ( A ) λ μ ( B ) 1 − λ , {\displaystyle \mu (\lambda A+(1-\lambda )B)\geq \mu (A)^{\lambda }\mu (B)^{1-\lambda },}

i.e.

log ⁡ μ ( λ A + ( 1 − λ ) B ) ≥ λ log ⁡ μ ( A ) + ( 1 − λ ) log ⁡ μ ( B ) . {\displaystyle \log \mu (\lambda A+(1-\lambda )B)\geq \lambda \log \mu (A)+(1-\lambda )\log \mu (B).}

Thus, a measure being 0-convex is the same thing as it being a logarithmically concave measure.

Properties

The classes of s-convex measures form a nested increasing family as s decreases to −∞"

s ≤ t  and  μ  is  t -convex ⟹ μ  is  s -convex {\displaystyle s\leq t{\text{ and }}\mu {\text{ is }}t{\text{-convex}}\implies \mu {\text{ is }}s{\text{-convex}}}

or, equivalently

s ≤ t ⟹ { s -convex measures } ⊇ { t -convex measures } . {\displaystyle s\leq t\implies \{s{\text{-convex measures}}\}\supseteq \{t{\text{-convex measures}}\}.}

Thus, the collection of −∞-convex measures is the largest such class, whereas the 0-convex measures (the logarithmically concave measures) are the smallest class.

The convexity of a measure μ on n-dimensional Euclidean space Rn in the sense above is closely related to the convexity of its probability density function.⁴ Indeed, μ is s-convex if and only if there is an absolutely continuous measure ν with probability density function ρ on some Rk so that μ is the push-forward on ν under a linear or affine map and e s , k ∘ ρ : R k → R {\displaystyle e_{s,k}\circ \rho \colon \mathbb {R} ^{k}\to \mathbb {R} } is a convex function, where

e s , k ( t ) = { t s / ( 1 − s k ) if  − ∞ < s < 0 t − 1 / k if  s = − ∞ , − log ⁡ t if  s = 0. {\displaystyle e_{s,k}(t)={\begin{cases}t^{s/(1-sk)}&{\text{if }}-\infty <s<0\\t^{-1/k}&{\text{if }}s=-\infty ,\\-\log t&{\text{if }}s=0.\end{cases}}}

Convex measures also satisfy a zero-one law: if G is a measurable additive subgroup of the vector space X (i.e. a measurable linear subspace), then the inner measure of G under μ,

μ ∗ ( G ) = sup { μ ( K ) ∣ K ⊆ G  and  K  is compact } , {\displaystyle \mu _{\ast }(G)=\sup\{\mu (K)\mid K\subseteq G{\text{ and }}K{\text{ is compact}}\},}

must be 0 or 1. (In the case that μ is a Radon measure, and hence inner regular, the measure μ and its inner measure coincide, so the μ-measure of G is then 0 or 1.)⁵

References

1. Borell, Christer (1974). "Convex measures on locally convex spaces". Ark. Mat. 12 (1–2): 239–252. doi:10.1007/BF02384761. ISSN 0004-2080. https://doi.org/10.1007%2FBF02384761 ↩

2. Borell, Christer (1975). "Convex set functions in d-space". Period. Math. Hungar. 6 (2): 111–136. doi:10.1007/BF02018814. ISSN 0031-5303. /wiki/Doi_(identifier) ↩

3. Borell, Christer (1974). "Convex measures on locally convex spaces". Ark. Mat. 12 (1–2): 239–252. doi:10.1007/BF02384761. ISSN 0004-2080. https://doi.org/10.1007%2FBF02384761 ↩

4. Borell, Christer (1975). "Convex set functions in d-space". Period. Math. Hungar. 6 (2): 111–136. doi:10.1007/BF02018814. ISSN 0031-5303. /wiki/Doi_(identifier) ↩

5. Borell, Christer (1974). "Convex measures on locally convex spaces". Ark. Mat. 12 (1–2): 239–252. doi:10.1007/BF02384761. ISSN 0004-2080. https://doi.org/10.1007%2FBF02384761 ↩