Is this definition already been defined?

Question

I have made a definition of s-convex set for some $s\in(0,1]$. The definition is: For a fixed $s\in(0,1]$, a subset $A\in\mathbb{R}^2$ is called s-convex if $$t^sx+(1-t)^sy\in A$$ for all $x,y\in A$ and $0\leq t\leq 1$. I don't know if this definition has already been defined. Please give any suggestion.

Answer 1

As the comments say, yes: this $s$-convexity is well-defined. Perhaps you're more interested in knowing how it differs from ordinary convexity.

For $s\neq1$ and nontrivial $t$, an $s$-combination of points $t^sx+(1-t)^sy$ is not an affine combination, i.e. the coefficients don't sum to $1$. So $s$-convexity really requires us to work in a vector space, not just an affine space. Maybe this is obvious, but I wanted to point out the consequence that an affine transformation of an $s$-convex set is not necessarily $s$-convex. On the contrary, if a proper subset $A\subset \mathbb R^2$ is $s$-convex, then I suspect that no nontrivial translation of $A$ is $s$-convex. It would be interesting to flesh out that conjecture.

Single points are not $s$-convex. Instead, the $s$-convex hull of a point $x\neq 0$ is the ray $[1,\infty)\cdot x$.

The convex hull of a pair of points is not generally a curve between them. For example, the $\frac12$-convex hull of the points $\{(1,0),(0,1)\}$ is the first quadrant of the plane, $[0,\infty)\times[0,\infty)$, minus the unit open ball.

The preceding example shows that an $s$-convex set need not be convex, and a convex set need not be $s$-convex. This is somewhat unfortunate; ideally when you set up a range of conditions like "$\alpha$-foo" sets, then you want every $\alpha$-foo set to be $\beta$-foo whenever $\alpha\geq\beta$. Or, when $\alpha\leq\beta$. Especially, if there's already a concept of a "foo set", then the $\alpha$-foo should be either stronger or weaker. It's nice to have some relationship.

Note in both the above examples that the $s$-convex hull of a finite set of points is not bounded, and it is not a polygon(al curve) unless you allow points on the line at infinity.

I haven't even thought about how $s$-convex functions will work. You'll probably have to be pretty careful while developing them.