Two functions Δ and Δ_{$b$}, of interest in combinatorial geometry and the theory of linear programming, are defined and studied. Δ($d, n$) is the maximum diameter of convex polyhedra of dimension$d$with$n$faces of dimension$d$−1; similarly, Δ_{$b$}($d,n$) is the maximum diameter of bounded polyhedra of dimension$d$with$n$faces of dimension$d$−1. The diameter of a polyhedron$P$is the smallest integer$l$such that any two vertices of$P$can be joined by a path of$l$or fewer edges of$P$. It is shown that the bounded$d$-step conjecture, i.e. Δ_{$b$}$(d,2d)=d$, is true for$d$≤5. It is also shown that the general$d$-step conjecture, i.e. Δ($d, 2d$)≤$d$, of significance in linear programming, is false for$d$≥4. A number of other specific values and bounds for Δ and Δ_{$b$}are presented.

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We shall be concerned with the indicator$p$of an analytic functional μ on a complex manifold$U$: $$p(\varphi ) = \overline {\mathop {\lim }\limits_{t \to + \infty } } \frac{l}{t}\log \left| {\mu (e^{t\varphi } )} \right|,$$ where ϕ is an arbitrary analytic function on$U$. More specifically, we shall consider the smallest upper semicontinuous majorant$p$^{$J$}of the restriction of$p$to a subspace £ of the analytic functions. An obvious problem is then to characterize the set of functions$p$^{$J$}which can occur as regularizations of indicators. In the case when$U$=$C$^{$n$}and £ is the space of all linear functions on$C$^{$n$}, this set can be described more easily as the set of functions(0.1) $$\mathop {\lim }\limits_{\theta \to \zeta } \overline {\mathop {\lim }\limits_{t \to + \infty } } \frac{l}{t}\log \left| {u(t\theta )} \right|$$ of$n$complex variables ζ∈$C$^{$n$}where$u$is an entire function of exponential type in$C$^{$n$}. We hall prove that a function in$C$^{$n$}is of the form (0.1) for some entire function$u$of exponential type if and only if it is plurisubharmonic and positively homogeneous of order one (Theorem 3.4). The proof is based on the characterization given by Fujita and Takeuchi of those open subsets of complex projective$n$-space which are Stein manifolds.

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