so i had succumbed to inward temptation today .. not only did i drink three servings of coffee [1] and eat a half-dozen brownies and several handfuls of roasted peas (an asian goodie), but i thought a little about mathematics today.
i had gone running, the thoughts wouldn't shut off [2], and somehow a theorem of Federer and Fleming came to mind: it states that the Sobolev Embedding Theorem for p = 1 gives rise to an isoperimetric inequality, and vice versa; better yet, the optimal constants are the same. from the pedestrian viewpoint [3] it does give a definitive correspondence between function theory and geometry, which is still not obvious to me.
the thought and a few fancies about Federer-Fleming lasted the three miles i ran .. not very long and not very deep thoughts, but they linger still ..
[1] .. one of these servings arose from breakfast at the diner. there the coffee was weak, but i had 2-3 refills of the stuff. i suppose that it evens out, then.
[2] .. after running cross-country and track in school, i can't not pay attention to minute details when running. listening for biofeedback and watching for race opponents has been ingrained in me over 5-6 years of youth competitive running, and now i can't turn it off: i can't stop thinking or processing sensory data when i run.
[3] .. assuming that this pedestrian had a little knowledge of Sobolev spaces and Hausdorff measures, that is. q:
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