Higher Genus
Whether or not the inequality
is satisfied by all surfaces of nonpositive Euler characteristic is unknown. For orientable surfaces of genus 2 and genus 20 and above, the answer is affirmative, see work by Katz and Sabourau below.
Read more about this topic: Loewner's Torus Inequality
Famous quotes containing the words higher and/or genus:
“Wherever a man separates from the multitude, and goes his own way in this mood, there indeed is a fork in the road, though ordinary travelers may see only a gap in the paling. His solitary path across lots will turn out the higher way of the two.”
—Henry David Thoreau (1817–1862)
“Methinks it would be some advantage to philosophy if men were named merely in the gross, as they are known. It would be necessary only to know the genus and perhaps the race or variety, to know the individual. We are not prepared to believe that every private soldier in a Roman army had a name of his own,—because we have not supposed that he had a character of his own.”
—Henry David Thoreau (1817–1862)