# Can we say something about the number of local maxima of this function ? Can we prove that it has at most $n−1$ local maxima?

Let $A\subseteq \mathbb{R}^n$ be the region defined with

$$A = \Big\{ (x_1,\ldots,x_n)\in\mathbb{R}^n \mid 0 \le x_i < 1\text{ for all i, and } \sum x_i < 1 \Big\}.$$

Let $\mathbf{u} =(u_1,\ldots,u_n)$ be an arbitrary unit vector with $u_i>0$ for all $i$.

Let $f: A\to \mathbb{R}$ be a multivariate function. And now, define $f_{\mathbf{u}}(t) = f(t\mathbf{u})$. Clearly, this is just the multivariate function $f$ going in the direction of the unit vector $\mathbf{u}$.

Denote as usual with $\mathbf{e}_i$, for some $i=1,\ldots,n$, the basis unit vectors, e.g. $\mathbf{e}_1 = (1,0,\ldots,0)$.

We assume that

1. $f_{\mathbf{u}}'(0) = u_1f_{\mathbf{e}_1}'(0) + \cdots + u_nf_{\mathbf{e}_n}'(0)$, i.e. that the derivative of $f$ at $0$ in any direction is just a linear combination of the derivatives in the basis directions.

2. $f_{\mathbf{u}}''(t) < 0$, i.e. that the derivative $f_{\mathbf{u}}'(t)$ is strictly decreasing. In other words: The slope of $f_{\mathbf{u}}(t)$ is decreasing as $t$ increases.

3. $f_{\mathbf{u}}(t) \to -\infty$ as $t\to 1/\sum u_i$ for any direction $\mathbf{u}$. In other words: In any direction, the multivariate function $f$ approaches minus infinity as the boundary of the region is approached.

My questions are these: Can we say something interesting about $f$ in general? In particular: Can we say something about the number of local maxima it has? Can we prove that it has at most $n-1$ local maxima? For $n=2$, it seems intuitively clear that it can have only 1 maximum, and I was wondering whether it would be possible to generalize this to $n$ dimensions.

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