Oct 22, 2024 · A function is convex if every point on the graph has a supporting line touching it. If the function is differentiable at a point , the supporting line is unique and I equal to the tangent line. …

Aug 16, 2019 · But if I have to check if a given function is convex or not, this definition seems hard and impractical to use. So, my question is, is there any easier way of checking convexity of a function …

A function $f$ is convex on an interval, if for all $a$ and $b$ in the interval, the line segment joining $ (a, f (a))$ and $ (b, f (b))$ lies above the graph of $f$.

Mar 27, 2015 · The function $g (x)$ is a concave. You can see from your graph that the line passing through two given points on the curve lies below the graph of $g$, not above the graph (which you …

The authors prove the proposition that every proper convex function defined on a finite-dimensional separated topological linear space is continuous on the interior of its effective domain. You can likely …

The epigraph of a convex $\mathbb {R} \to \mathbb {R}$ function is a convex set within ``graph space'' $ (x,y)$ although this doesn't explain why we should look at the epigraph instead of the subgraph….

Further comment on the intuition of convexity was given by Raskolnikov, who said Replace the inequality by an equality and you get a linear function. Thus, you could say that convexity is a form of …

Oct 19, 2018 · Yes, if $E$ is an infinite-dimensional real Banach space then a discontinuous linear functional is a discontinuous convex function. But the map $f$ defined by $f (u)=\sum u_i/i$ is …

A strongly convex function is strictly convex but the converse need not be true. The condition for strict convexity is strict Jensen's inequality as pointed out by Alex R.

Besides using the (zero-th order) definition of function convexity (chord is above graph), we can prove the convexity of its epigraph S, which is iff condition of function convexity.