mostly From economical viewpoint taking f as

From economical viewpoint, taking f as a production cost, as a production supply and as a vector price. Here, conjugate function f* represent maximum profile in the production. Unfortunately, these economical interpretations hold when f is convex and lower semi continuous, because defined by is the closed convexication of f (f**=f). When f is no convex we loss all these economical interpretations because f**≠f. The good news is that there exists in the literature extensions of Fenchel\'s conjugate, for instance we have Fenchel–Moreau conjugate for lower semi continuous functions introduced in Cotrina et al. (2011), defined bywhere is a lower semi continuous function and is a continuous operator. So, when f is proper and is the continuous operator space. Following Cotrina et al. (2011), two functions and are conjugate functions if and only if g=f* and f=f**. Moreover a linear subspace which contains all constant operators of , is a dual conjugate space for f if and only if
Existence result for the consumer problem Here we introduce a characterization of the solution set for the consumer problem (PC), using Fenchel–Moreau conjugate.If we define by (see previous section for more details), the following results establish a necessary and sufficient optimality condition for this general model of consumer problem (without assumptions for both u and BS).
The following results reveal that PC has at least one solution if BS is a compact and u is upper closed.
For each and consider . So, given consider .
For example, consider defined by p(x):=Ax, where A is a nonsingular, symmetric and positive definite matrix (hence, a linear monotone operator) and such that 〈p(ω), ω〉>0, we have that for any upper closed utility function and BS=BS(p, ω), the respective Consumer problem has at least one solution, since Theorem 2. As a consequence of Theorem 2, we obtain the following result, which improve Weierstrass Theorem (remember that the family of closed upper functions contains the family of upper semi-continuous functions, but both mostly are different). Similar result you can find in Aliprantis and Border (1999), Theorem 2.43.
When BS is unbounded, we consider the following assumption: From optimization viewpoint, assumption WS, can be consider as a coercive condition. For more detailed see Sosa (2013) and references there in. From consumer problem, assumption WS establish that for any sequence of choices x ⊂BS going to +∞ in measure, always there exist an element x of the sequence and another one choice such that in measure is strictly less than x and .
As a consequence of Theorem 3 we obtain the following result, adopting assumption WS for any function and any set . We denote this assumption by WS(u,C).
We finished this section with an academical example in order to understand condition WS. Consider PC where u is a Cobb–Douglas function defined by , and . Of course BS(p, ω) is unbounded. By applying Theorem 3, PC has solution if and only if WS holds. The question is: What does mean WS in this academical example?. The answer is: If BS(p, ω)∩dom(u)≠∅, then WS is equivalent to p∈dom(u). Of course, the reader can verify it without difficulties.
Acknowledgements Wilfredo Sosa was partially supported by CNPq Grants 302074/2012-0 and 471168/2013-0. Part of this research was carried out during Wilfredo\'s visit to IMPA of Rio de Janeiro.
Introduction Victimization has been a major issue in Brazilian society in recent years. About 10% of the Brazilian population over 10 years old had been victimized by at least one crime between September 2008 and September 2009 (PNAD, 2009). Of these, 5.4% were victims of attempted theft or robbery, 1.5% were physically assaulted, 3.7% were robbed, and 3.9% were victims of theft. In addition, 51% of all people aged 10 or over felt insecure. The fear of being victimized reduces the well-being of individuals and, consequently, the social well-being. It is therefore important to investigate what are the “determinants” of victimization. Routine activities outside one\'s home increase the risk of victimization and convergence in time and space between victims, criminals, and the absence of police leads to victimization even without changes in the variables that determine criminal behavior (Cohen and Felson, 1979). Based on this concept, Cohen et al. (1981) formalized a “theory” according to which the risk of victimization is determined by five factors: exposure, guardianship, proximity to potential offenders, attractiveness of potential targets, and definitional properties of specific crimes themselves. The first three factors are related to routine and the fourth one is related to the victim\'s attributes, especially economic attributes.