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 Apr10 awarded Yearling Sep30 awarded Explainer Sep24 awarded Autobiographer Sep15 awarded Self-Learner Aug9 reviewed Approve How to solve this IVP? Aug9 comment Do the radii of a family of nested balls (in a Banach space) converge? That's interesting and surprising for me, many thanks. Can I ask you which is the problem for $\mathbb Q_p$, thus? Why does Henning Makholm's proof below does not apply in thi case? I am not very familiar with $p$-adic numbers, to be honest. Thanks again for your comments. Aug9 comment Do the radii of a family of nested balls (in a Banach space) converge? @you-sir-33433 Sure, thanks for pointing it out; but if you are in a Banach space and the balls are nested then necessarily the intersection is not empty. Aug9 comment Outer Measure is not Finite Additive You may find this interesting. math.stackexchange.com/questions/878962/… Aug9 accepted Do the radii of a family of nested balls (in a Banach space) converge? Aug9 comment Do the radii of a family of nested balls (in a Banach space) converge? Oh, I see your point. Thank you very much for the kind and fast reply. Aug9 asked Do the radii of a family of nested balls (in a Banach space) converge? Aug9 comment Nested sequences of balls in a Banach space Sorry for this comment which is probably stupid but I am wondering how you would prove that $B_r(x) \subset B_s(x)$ implies $r \le s$. What about these examples: math.stackexchange.com/questions/734248/… ? Thanks. Aug8 accepted A strictly convex function defines an implicit function with non-positive second derivative Aug7 revised Jordan Measure and Lebesgue Measure edited title Aug6 comment How do I approach the problems asking about uniqueness @user86418 Oh, sorry: you are perfectly right, I have been stupid. Thanks for pointing it out, now it makes sense. Aug6 comment How do I approach the problems asking about uniqueness The solution of the first problem is straigthforward if we do not ask for periodicity (just take $C(\mathbb R)$ with the sup norm and verify that the "obvius" map is 1/2-Lipschitz: then apply Banach-Caccioppoli fixed point theorem). I am wondering what changes adding periodicity condition: the space of continuous periodic functions is still complete under the sup norm hence I think that the same argument applies. Aug5 revised A strictly convex function defines an implicit function with non-positive second derivative Added a possible solution to new doubt Aug5 revised A strictly convex function defines an implicit function with non-positive second derivative Added a new doubt Aug4 comment A strictly convex function defines an implicit function with non-positive second derivative This is nice, thank you for your answer. I have been very stupid because I did not notice that. Thanks again. By the way, what do you think about the first point? Is it correct? I am now doubtful because I have proved only that the difference quotients are $>0$, maybe when we pass to the limit we get $0$... Thanks. Aug4 asked A strictly convex function defines an implicit function with non-positive second derivative