Satoshi Murai and Eran Nevo have just proved the 1971 generalized lower bound conjecture of McMullen and Walkup, in their paper *On the generalized lower bound conjecture for polytopes and spheres* . Let me tell you a little about it. For more background see the post: How the g-conjecture came about.

**Face numbers and h-numbers**

Let *P* be a *(d-1)-*dimensional simplicial polytope and let be the number of -dimensional faces of P. The -**vector** (face vector) of P is the vector .

Face numbers of simplicial *d*-polytopes are nicely expressed via certain linear combinations called the* h*-numbers. Those are defined by the relation:

What’s called “Stanley’s trick” is a convenient way to practically compute one from the other, as illustrated in the difference table below, taken from Ziegler’s book `Lectures on Polytopes’, p.251:

**1**

1 **6**

1 5 **12**

1 4 7 ** 8**

h= (1 3 3 1)

Here, we start with the -vector of the Octahedron (1,6,12,8) (bold face entries) and take differences as shown in this picture to end with the -vector (1,3,3,1).

The **Euler-Poincare** relation asserts that . More is true. The **Dehn-Sommerville** relations state that is symmetric, i.e. for every .

**The generalized lower bound conjecture**

In 1971, McMullen and Walkup posed the following conjecture, which is called the generalized lower bound conjecture (GLBC):

Let *P* be a simplicial *d*-polytope. Then

(A) the

h-vector ofP,satisfies .(B) If for some then can be triangulated without introducing simplices of dimension .

The first part of the conjecture was solved by Stanley in 1980 using the Hard Lefschetz theorem for toric varieties. This was part of the *g*-theorem that we discussed extensively in a series of posts (I, I’, II, III, B). In their paper, Murai and Nevo give a proof of part (B). This is remarkable!

Earlier posts on the g-conjecture:

I: (Eran Nevo) The g-conjecture I

I’ How the g-conjecture came about

II (Eran Nevo) The g-conjecture II: The commutative-algebra connection

III (Eran Nevo) The g-conjecture III: Algebraic shifting

B: Billerafest

Congratulations to Eran and Satoshi!

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