Intereting Posts

A triangle determinant that is always zero
How to find $\lim_{x \to \infty} /x$?
Show $T$ is invertible if $T'$ is invertible where $T\in B(X)$, $T'\in B(X')$
One-to-one function from the interval $$ to $\mathbb{R}\setminus\mathbb{Q}$?
Are variables logical or non-logical symbols in a logic system?
If $(A-B)^2=O_2$ then $\det(A^2 – B^2)=(\det(A) – \det(B))^2$
Probability distribution of the maximum of random variables
Resolvent: Definition
Unambiguous terminology for domains, ranges, sources and targets.
Is it true that the unit ball is compact in a normed linear space iff the space is finite-dimensional?
Example of a linear operator on some vector space with more than one right inverse.
The set of all functions from $\mathbb{N} \to \{0, 1\}$ is uncountable?
basis of a vector space
PDF of a sum of exponential random variables
How was the difference of the Fransén–Robinson constant and Euler's number found?

Let $f(t)$ be a convex function and define $g(t)$ to be the running average of $f(t)$

$$g(t) = \frac{1}{t} \displaystyle\int_0^t f(\tau) ~d\tau$$ Then $g$ is convex.

This is easy enough to prove just by differentiating twice. My question is: is there an easy geometric proof of this statement?

- mid-point convex but not a.e. equal to a convex function
- $f$ is convex function iff Hessian matrix is nonnegative-definite.
- Hausdorff metric and convex hull
- If $f$ is midpoint convex, continuous, and two times differentiable, then for any $a, b \in \mathbb{R}$, there exists $c$ such that $f''(c) \geq 0$
- A problem about convex domains
- Hessian matrix for convexity of multidimensional function

- Volume of the projection of the unit cube on a hyperplane
- An explicit construction for an everywhere discontinuous real function with $F((a+b)/2)\leq(F(a)+F(b))/2$?
- Extreme points of the unit ball of the space $c_0 = \{ \{x_n\}_{n=1}^\infty \in \ell^\infty : \lim_{n\to\infty} x_n = 0\}$
- Proof of convexity of $f(x)=x^2$
- Proof that the intersection of any finite number of convex sets is a convex set
- Mathematical expectation is inside convex hull of support
- how to show $f$ attains a minimum?
- Show that $f(x)=||x||^p, p\ge 1$ is convex function on $\mathbb{R}^n$
- Convexity of the product of two functions in higher dimensions
- Convex hull of orthogonal matrices

I would not differentiate twice, only because of the simple reason that you don’t even know that $f$ is differentiable. However, there is a geometric way of looking at this. A convex function $f$ is defined by the following property: the graph of $f(x)$ between $x = a$ and $x = b$ lies below the line connecting the points $(a,f(a))$, $(b, f(b))$. I would like to say that pictorially I can think of the averages of the endpoints as always being larger than the full average over the whole interval $(a,b)$. So when you’re averaging the average, this should remain the case. But maybe that isn’t really precise enough…

Sean Eberhard’s change of variables is very useful for formalizing this; for $r \in [0,1]$,

$$ rg(a) + (1-r)g(b) = \int_0^1 rf(as) + (1 -r)f(bs) \, ds \geq \int_0^1 f(ras + (1-r)bs) \, ds = g(ra + (1 – r)b). $$

- Mathematical Induction
- Find the equation of the plane passing through a point and a vector orthogonal
- Power residue theorem without p-adic method
- Composition of a weakly convergent sequence with a nonlinear function
- Prove that $\sqrt5 – \sqrt3$ is Irrational
- Is conjugate of holomorphic function holomorphic?
- Show that $\mathbb{Q}$ is dense in the real numbers. (Using Supremum)
- Infinite sum involving ascending powers
- Density of positive multiples of an irrational number
- Sum of the first $n$ triangular numbers – induction
- Prove a graph with $2n-2$ edges has two cycles of equal length
- prove: A finitely generated abelian group can not be isomorphic to a proper quotient group of itself.
- Does the rank of homology and cohomology groups always coincide?
- Infinite Prime Numbers: With Fermat Numbers
- Is there a Maximum Principle for Biharmonic eigenvalue problem?