Boros and Füredi (for d = 2) and Bárány (for arbitrary d) proved that there exists a positive real number cd such that for every set P of n points in R d in general position, there exists a point of R contained in at least cd ( n d+1 ) d-simplices with vertices at the points of P . Gromov improved the known lower bound on cd by topological means. Using methods from extremal combinatorics, we im...

A stabbing line for a set of convex polyhedra is extremal if it passes through four edges and is tangent to the polyhedra containing those edges. In this paper we present three constructions of convex polyhedra with many extremal lines. The rst construction shows (n 2) extremal stabbing lines constrained to meet two skew lines. The second shows (n 4) extremal lines which are tangent to two poly...

We give several pieces of evidence to show that extremal black holes cannot be obtained as limits of non-extremal black holes. We review arguments in the literature showing that the entropy of extremal black holes is zero, while that of near-extremal ones obey the Bekenstein-Hawking formula. However, from the counting of degeneracy of quantum (BPS) states of string theory the entropy of extrema...

In this paper, new binary extremal self-dual codes are presented. A number of new extremal singly-even self-dual codes of lengths 48; 64 and 78, and extremal doubly-even self-dual codes of lengths 80 and 88, are constructed. We also relate an extremal doubly-even self-dual code of length divisible by 24 to an extremal singly-even self-dual code of that length. New singly-even self-dual codes of...

We show that certain geometrically defined higher codimension cycles are extremal in the effective cone of the moduli spaceMg,n of stable genus g curves with n ordered marked points. In particular, we prove that codimension two boundary strata are extremal and exhibit extremal boundary strata of higher codimension. We also show that the locus of hyperelliptic curves with a marked Weierstrass po...

In this work, quadratic double and quadratic bordered double circulant constructions are applied to F4 + uF4 as well as F4, as a result of which extremal binary self-dual codes of length 56 and 64 are obtained. The binary extension theorems as well as the ring extension version are used to obtain 7 extremal self-dual binary codes of length 58, 24 extremal self-dual binary codes of length 66 and...

We prove uniqueness theorems for asymptotically flat, stationary, extremal, vacuum black hole solutions, in four and five dimensions with one and two commuting rotational Killing fields respectively. As in the non-extremal case, these problems may be cast as boundary value problems on the two dimensional orbit space. We show that the orbit space for solutions with one extremal horizon is homeom...

Motivated by the recently proposed Kerr/CFT correspondence, we investigate the holographic dual of the extremal and non-extremal rotating linear dilaton black hole in Einstein-Maxwell-Dilaton-Axion Gravity. For the case of extremal black hole, by imposing the appropriate boundary condition at spatial infinity of the near horizon extremal geometry, the Virasoro algebra of conserved charges assoc...

This paper presents explicit equations over Q for 32 extremal semistable elliptic K3 surfaces. They are realized as pull-back of non-semi-stable extremal rational elliptic surfaces via base change. Together with work of J. Top and N. Yui which exhibited the same procedure for the semi-stable extremal rational elliptic surfaces, this exhausts this approach to produce extremal semi-stable ellipti...