Dunkl-Williams inequality

Let $V$ be an inner product space and $a,b\in V$. If $a\ne 0$ and $b\ne 0$, then

$$\parallel a-b\parallel \ge \frac{1}{2}(\parallel a\parallel +\parallel b\parallel )\parallel \frac{a}{\parallel a\parallel }-\frac{b}{\parallel b\parallel }\parallel .$$(1)

Equality holds if and only if $a=0$, $b=0$, $\parallel a\parallel =\parallel b\parallel$or $a\parallel b\parallel =b\parallel a\parallel$.In fact, if (1) holdsand $V$ is a normed linear space, then $V$ is an inner product space.

If $X$ is a normed linear space and $a\ne 0$ and $b\ne 0$ then

$$\parallel a-b\parallel \ge \frac{1}{4}(\parallel a\parallel +\parallel b\parallel )\parallel \frac{a}{\parallel a\parallel }-\frac{b}{\parallel b\parallel }\parallel .$$(2)

Equality holds if and only if $a=0$, $b=0$ or $a=b$.The constant $\frac{1}{4}$ is best possible. For example, let $X$ be the set of ordered pairs of real numbers,with norm of $(x_1,x_2)$ equal to $|x_1|+|x_2|.$ Let$a=(1,ϵ)$ and $b=(1,0)$ where $ϵ$ is a small positive number. After a bitof routine calculation, it is easily seen that the best possible constant is $\frac{1}{4}$.

The inequality (2) has been generalized in the case where $X$ is a normed linear spaceover the reals. In that case one can show:

$$\parallel a-b\parallel \ge c_p{({\parallel a\parallel }^p+{\parallel b\parallel }^p)}^{1/p}\parallel \frac{a}{\parallel a\parallel }-\frac{b}{\parallel b\parallel }\parallel$$(3)

where $c_p=2^{-1-1/p}$ if and $c_p=1/4$ if $p\ge 1$. The case $p=1$ is theDunkl and Williams inequality.

If $X$ is a normed linear space and then (3) holds with $c_p=2^{-1/p}$if and only if $X$ is an inner product space.

The inequality (2) can be improved slightly to get:

$$\parallel a-b\parallel \ge \frac{1}{2}\max (\parallel a\parallel ,\parallel b\parallel )\parallel \frac{a}{\parallel a\parallel }-\frac{b}{\parallel b\parallel }\parallel .$$(4)

Equality holds in (4) if and only if $a$ and $b$ span an $\mathrm{\ell }_2{}^1$ in the underlying real vector space with$\pm {\parallel b-a\parallel }^{-1}(b-a)$ and $\pm {\parallel a\parallel }^{-1}a$ (or $\pm {\parallel b\parallel }^{-1}b$) as the vertices of the unitparallelogram.