我的英文答案如下：

1、Every lower contour set of the vector space of p,w is convex, i.e. all sets {(p,w):v(p,w)<=u} are convex for each utility level u. As a result, the indirect utility function is quasi-convex.

For n-dimension Euclidean space, we have theorem that: for f: R^k->R (a mapping from k-dimension vector whose elements are real numbers to real number), that every upper contour set of the domain is convex is equivalent to that the function is quasi-concave, that every lower contour set of the domain is convex is equivalent to that the function is quasi-convex.

The reason why every lower contour set is convex is due to the convexity of the preference relation. For any different price-wealth pair (p,w) and (p',w'), we can get two different optimal bundles through the demand function x(p,w), and they are both points given by the intersection of the convex indifference curves and the budget line. The new price-wealth (ap+(1-a)p,aw+(1-a)w) draws a new budget line tilted around the intersection of the original two budget lines. We can easily find that the new utility level given by the new price-wealth pair is no more than the maxmum utility of the original price-wealth pair.That is, for any two different price-wealth pair (p,w) and (p',w'), we have v(ap+(1-a)p,aw+(1-a)w) <= max[v(p,w),v(p',w')].

The convexity of the preference determines the the quasi-convexity of the indirect utility function. Another crucial assumption is the local nonsatiation, which says that we can always find a more preferred point around a certain point on consumption bundle space. This assumption rules out the extreme situation in which the consumer prefers a certain kind of comsumption budle to any other bundles. And the local nonsatiation is the basic assumption of the Walras Law.

2.The monotonicity of the preference relation means the utility function is monotone, i.e. for any x and y in R^k, y >> x implies y is preferred to x. (y>>x means each element of y is greater than the corresponding element of x). Utility function is a mapping from n-dimension vector to a real number, and it actually assigns numbers to different consumption bundles in an order-preserving way.

其实大概只要用到微积分的知识就可以解决了

小弟也在拜读MAS-COLELL的那本书

其实那个是有答案的

你在网上找一找吧

然后很多问题就很好解决了