Results 1 
1 of
1
On the SelfConcordance of the Universal Barrier Function
, 1995
"... Let K be a regular convex cone in R n and F (x) its universal barrier function. Let D k F (x)[h; : : : ; h] be kth order directional derivative at the point x 2 K 0 and direction h 2 R n . We show that for every m 3 there exists a constant c(m) ? 0 depending only on m such that jD m F (x) ..."
Abstract

Cited by 6 (1 self)
 Add to MetaCart
Let K be a regular convex cone in R n and F (x) its universal barrier function. Let D k F (x)[h; : : : ; h] be kth order directional derivative at the point x 2 K 0 and direction h 2 R n . We show that for every m 3 there exists a constant c(m) ? 0 depending only on m such that jD m F (x)[h; : : : ; h]j c(m) D 2 F (x)[h; h] m=2 . For m = 3, this is the selfconcordance inequality of Nesterov and Nemirovskii. Our proof uses a powerful recent result of Bourgain.