Stochastic Programming - Index of

BASIC CONCEPTS 51
and hence
P (λS 1 +(1− λ)S 2 ) ≥ min[P (S 1 ),P(S 2 )].
As mentioned above, for the log-concave case necessary and sufficient
conditions were derived first, and later corresponding conditions for quasiconcave
measures were found.
Proposition 1.6 Let P on Ξ=IR k be of the continuous type, i.e. have a
density f. Then the following statements hold:
• P is log-concave iff f is log-concave (i.e. if the logarithm of f is a concave
function);
• P is quasi-concave iff f −1/k is convex.
The proof has to be omitted here, since it would require a rather advanced
knowledge of measure theory.
Remark 1.4 Consider
(a) the k-dimensional uniform distribution on a convex body S ⊂ IR k (with
positive natural { measure µ) given by the density
1/µ(S) if x ∈ S,
ϕ U (x) :=
0 otherwise
(µ is the natural measure in IR k , see Section 1.4.1);
(b) the exponential { distribution with density
0 if x0 is constant);
(c) the multivariate normal distribution in IR k described by the density
ϕ N (x) :=γe − 1 2 (x−m)T Σ −1 (x−m)
(γ > 0 is constant, m is the vector of expected values and Σ is the
covariance matrix).
Then we get immediately
{ √
k
(a) ϕ − 1 k
µ(S) if x ∈ S,
U (x) = ∞ otherwise,
implying by Proposition 1.6 that the corresponding propability measure
P U is quasi-concave.
✷