Let $X$ be a vector space. The positive-homogeneous function $\|\cdot\|$ is said to be a quasinorm if $\|x+y\|\le K(\|x\|+\|y\|)$, for some $K\ge1$; it is a norm if $K=1$.
Question: 1. (terminology) if we drop the positive-homogeneity assumption and only assume that $f:X\to\mathbb{R}$ is continuous and satisfies $f(\frac12x+\frac12y)\le \frac{K}{2}[f(x)+f(y)]$, one might expect, by analogy, that this property be called "quasiconvexity" but it is not; the latter term is reserved for $f(\frac12x+\frac12y)\le \max\{f(x),f(y)\}$. Q: So what is the accepted term for this property? 2. What is known about maximizing a quasinorm (or, more generally, a "weakly convex" function $f$ in the sense of Q1) over a compact set -- say, in a finite-dimensional space? Are the maxima necessarily achieved at extreme points?
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