How to prove that: $\forall a \in \Bbb R,$ $\|f+a\|_p\ge \frac12\|f\|_p$ when $\int_\Omega f(x)dx= 0$ Let $\Omega $ be any subset of $\Bbb R^d$ and $f\in L^p(\Omega)$ such that 
$$\int_\Omega f(x)dx= 0$$ 

Then, shows that for each real number  $a\in \Bbb R,$ we have $$\|f+a\|_{L^p(\Omega )}\ge \frac12\|f\|_{L^p(\Omega )}$$

This looks obvious but but spent already a lot of time on it. 
I noticed that it sufficed to prove the inequality  for $a>0 $. the case  $p=2$ is been trivial using quadratic.
Any Hint or idea?
 A: You need to assume $a\in L^p(\Omega)$, otherwise $f+a\notin L^p(\Omega)$. By Minkowski's inequality, $$\|f\|_p\leq \|f+a\|_p+\|a\|_p $$
Now it remains to prove that $\|a\|_p\leq \|f+a\|_p $, using the fact that $\int_{\Omega}f=0$. 
Hint: Apply Jensen's inequality to $f+a$ and $x\mapsto |x|^p$.
Edit: I suppose you can also obtain it by applying Holder's inequality to $(f+a)\cdot 1$.
Edit2: I will write a more detailed proof since it is not clear enough apparently. OP is considering a function $f+a$, where $f\in L^p(\Omega)$ and $a\in \mathbb{R}$. The function $f+a$ is the map
$$ (f+a)(x)= f(x)+a,\qquad \forall x\in \Omega$$
This is the exactly the same as considering the sum of functions $f+a$, where $a$ is the constant function $a(x)=a$ for all $x\in \Omega$. Notice that
$$(f+a)(x)=f(x)+a(x)=f(x)+a,\qquad \forall x\in \Omega $$
which is exactly the equality written above. As a function, $a\in L^p(\Omega)$, i.e. $\|a\|_p<\infty$, because $L^p(\Omega)$ is a vector space and $f+a,f\in L^p(\Omega)$. Hence by Minkowski's inequality, 
$$\|f\|_p\leq \|f+a\|_p+\|a\|_p $$
Assume $p\neq \infty$:
since $a\in L^p$, we have $\mu(\Omega)<\infty$. By applying Holder's inequality to $1\cdot(f+a)$ we obtain
$$\mu(\Omega)|a|=\left|\int_{\Omega} a\right| =\left|\int_{\Omega}(f+a)\right|\leq \int_{\Omega}|f+a|\leq \|1\|_{p'}\|f+a\|_p=\mu(\Omega)^{1-1/p}\|f+a\|_p$$
Thus
$$\|a\|_p=\left(\int_{\Omega}|a|^p\right)^{1/p}=\mu(\Omega)^{1/p}|a|\leq\|f+a\|_p $$
If $p=\infty$ the inequality $\|a\|_{\infty}\leq \|f+a\|_{\infty}$ is trivial because since $\int_{\Omega}f=0$, then $f$ takes both non-negative and non-positive values on nonzero measure subsets of $\Omega$.
Putting this together with Minkowski's inequality we get $\|f\|_p\leq 2\|f+a\|_p$.
