in the above comment by an open bounded set in Rn we mean just a body in Rn...and that is given to be a convex body

Maybe convexity is not implied but in this reading basically we let X=Closure(K) where K is a convex body in Rn(i.e an open bounded set in Rn).Then Cl(K) is a compact convex set....and basically we want to show that r-dimensional Vol: Cl(K)->R is a continuous function. But the above question is stated as an exercise and is stated exactly like above

but the problem is stated exactly as above

Convexity i think is implied since L is a compact set of Rn....so convexity is part of the question

In what I am reading the metric topology of O(X) is used to construct some continuous functions(which are used in the proof of Minkowski's Second theorem). And seems like the Vol(L) should be a continuous function. Isn't Vol(L) the Lebesgue Measure of L? Since L is after all a compact set in Rn i think its volume is the Lebesgue Measure of L....and that is a number! So the volume must be a function V:O(X)->R

the metric is defined as d(L1,L2)=sup{d(x1,L2):x1 in L1}+sup{d(x2,L1):x2 in L2}. Here L1 and L2 are nonempty compact subsets of Rn and by distance from a vector x to the set L we mean d(x,L)=inf{d(x,y):y in L}. With this metric is not difficult to show that O(X) is a metric space but I don't know how to prove continuity of vol(L) on O(X)! Do I use the open set approach or delta-sigma def? Help! Thank you – Tanja 0 secs ago

the metric is defined as d(L1,L2)=sup{d(x1,L2):x1 in L1}+sup{d(x2,L1):x2 in L2}. Here L1 and L2 are nonempty compact subsets of Rn and by distance from a vector x to the set L we mean d(x,L)=inf{d(x,y):y in L}. With this metric is not difficult to show that O(X) is a metric space but I don't know how to prove continuity of vol(L) on O(X)! Do I use the open set approach or delta-sigma def? Help! Thank you