MATH . SCAND. 30 (1972), 297-312

A REPRESENTATION THEOREM FOR A CONVEX CONE OF QUASI CONVEX FUNCTIONS

S . JOHANSEN

0 . Introduction and summary.

It was conjectured by Br0ns [4] that if a family of quasi convex tions defined on R is closed under addition, then it could be transformed into a family of convex functions by a monotone transformation of the domaine of definition.

In this paper we give the proof of this conjecture and apply it to the problem of estimating a parameter from a one parameter statistical problem with unimodal likelihood function, see van Eeden [6].

A quasi convex function defined on R is just a function which is first decreasing and then increasing. The elementary properties are listed in section 1.

It should be noted that a sum of two quasi convex functions is in eral not quasi convex. The condition that a family of quasi convex tions is in fact closed under addition is thus a strong condition.

The family of convex functions on R is clearly closed under addition and one could ask whether a family of quasi convex functions closed under addition is infact a family of convex functions. This can not be true since quasi convexity is invariant under homeomorphic tions of R and convexity is not.

We can however, turn the problem around and ask if we can choose a transformation of R, depending on the family in question, such that in the new scale, the functions become convex. This is done in Theorem 2.8. Under differentiability conditions we can give an explicit form for the transformation. Theorem 2.4.

Since convex functions have derivatives whereas quasi convex tions need not, the condition on the family in fact implies that the tions considered as measures on R are equivalent.

In Section 3 we prove that for a family of quasi convex functions which is invariant under translation we can transform the domain of definition by a logarithm in order to get convexity.

Received July 28, 1971.