Any line intersect with $bd(S)$ at at most two points. Then $S$ or $S^c$ must be convex.
32110
$\forall x,y\in\partial S$, the line connecting $\{x,y\}$, $L\{x,y\}\cap\partial S=\{x,y\}$, then $S$ must be either convex or concave?
30572
Intersection of nested closed bounded convex sets in Euclidean space
73345
maximum size of a $k$-intersecting family of $[n]$
71074
The locus of the intersection point of two perpendicular tangents to a given ellipse
66151
Solving inequality for convex functions.
30423
interior points and convexity
116382
Showing convexity of a function with the restriction over an arbitrary line
124425
A geometry question: Let $ABCD$ be a convex quadrilateral. Prove that AD is parallel to BC.
72659
How can we define convexity in one dimension?
52231
Prove or disprove {$\sum_{i=1}^n x_i^2 = 1$} is convex
6335
Proving that the set enclosed by an ellipse is convex
57943
If $G$ is a convex set, prove that $\mathbb{C}\cup\{\infty\}\setminus G$ is connected.
50966
In a topological space, why the intersection only has to be finite?
48490
Examples of $f$ strictly convex, either with one minimizer or with no minimizer.
77217
Proof that a coordinate-wise convex function is convex?
102959
Show that $f$ is convex if and only if $f\left( \sum_{i=1}^m\lambda_ix_i \right) \leq \sum_{i=1}^m\lambda_if(x_i)$
56081
Prove that the ellipsoid $E = \{ (x,y,z) \in R^3 \mid \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} \leq 1 \}$ is convex