Since there's no need for secrecy regarding the party decorations any more I can post this shocking news on social inequality in Helsinki. Some of you have heard me promote equality for all balls. But now that I think about it I have to come to the conclusion that also people should be able to enjoy equality.
When planning Kirsi's party we decided that we wanted maple leaves as decoration since the theme (Yes, we had a theme!) was fall. The leaves would be placed on top of everybodys napkins on the tables. So they needed to be neat and pretty. But all the maple leaves in Pasila are full of ugly black spots. And so are the ones in Kumpula and in Käpylä and in basically every middle class neighbourhood in Helsinki. I was almost desperate for a while but then my mum suggested (mums are always right!) going to Kaivopuisto which is the most southern part of the Helsinki mainland where all the richest embassies and the fanciest apartment buildings are.
And it so happens that all the maple leaves in Kaivopuisto are big, beautiful, perfect and totally devoid of any annoying black spots or other defects. Now is that fair, I only ask? The leaders of the city should really look into this! It is totally unjust that the rich get not only more money but better parts of nature as well.
The pictures are from Kaivopuisto. No maple leaf pictures from Pasila since they would just be ugly.
6 comments:
Maple leaves are a good example of a covering family of sets. They are not disjoint, but since the surface of Earth is a doubling metric space, Exercises 2.10 and 2.11 from Juha's book can be used here.
Which reminds me: did you find an answer to the question about doubling measures and rectifiable curves? It's an interesting one.
actually, fallen leaves reminds me more of the besicovitch covering theorem on Euclidean n-space (the one about finite overlap).
was that the exercise you were referring to, L? i don't have my copy on me.
No, 2.10 says that if the sum of characteristic functions of balls B_i is in L^p, then the rescaled balls cB_i also have this property.
I think Blogger should support LaTeX. Even Facebook does.
I like the fact that the leaves are roundish like sets of any respectable covering should be. Namely that they contain a ball of radius r and are inside a ball of radius R with R/r bounded.
On doubling measures and rectifiable curves I'm sad to tell that I don't know the answer. We were working on the question last spring with Juha and we did find a fancy proof for our conjecture of the blue cubes for doubling measures with small doubling constants. A consequence is that all rectifiable curves have measure zero for all doubling measures with small doubling constants. This, however, turns out to be trivial. I'm happy to know it just the same though of course.
Sorry for not being more specific here but I'd get it wrong anyway and it's one thing to say stupid things and another altogether to have them on the Internet for everybody to read... See, I am self-concious!
Oh, and please feel free to point out that I've already written quite a many stupid things here.
facebook supports LaTeX?
(i only knew that wordpress supports LaTeX.)
i would be so quick to say "stupid." i don't know anything about the problem, but it doesn't sound easy.
The problem is definitely not easy, and considering how weird those non-sigma-porous rectifiable curves are, I will not be surprised if one of them is not null for some doubling measure.
The small constant case is indeed easy: a doubling measure satisfies mu(B(x,r))\le Cr^{\alpha} where alpha depends on the doubling constant. If the constant is small enough (in dimension 2 or higher) then alpha>1, hence rectifiable curves are null sets.
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