Given convex differentiable function $h: R^m to R$, what is geometric interpretation of $h(x)/|nabla h(x)|$ vis à vis the level-set $h=0$?

Question

Let $h:mathbb R^m to mathbb R$ be a convex diifferentiable function and define $C_h := {x in mathbb R^m mid h(x) = 0}$, assumed to be non-empty.

Question. Given $x in mathbb R^m$ such that $nabla h(x) ne 0$, is there any geometric interpretation of the quantity $omega_h(x) := (h(x))_+/|nabla h(x)|$ vis à vis the set $C_h$ ?

Motivation. In the linear case where $h(x) = v^Tx$ for some unit-vector $v in mathbb R^m$, one has

$C_h = v^perp$, the hyperplane perpendicular to $v$.
$|nabla h(x)| = |v| = 1$ for all $x in mathbb R^m$, and so $omega_h(x) = (h(x))_+ = (v^Tx)_+$, which is the distance of $x$ from $C_h$.

dohmatob · Answer

Given a unit-vector $v in mathbb S_{m-1} := {v in mathbb R^m mid |v| = 1}$, let $mathscr L_v := {x + tv mid
t ge 0} subseteq  mathbb R^m$ be the ray generated by $v$ and starting at $x$. The
distance of $x$ from $C_h$ can be written as
$$
r_h(x) := d(x,C_h) := inf_{x' in C_h}|x'-x| = inf_{v
  in mathbb S_{m-1}}inf_{x' in
  mathscr L_v cap C_h}|x'-x|.
$$
Now, by definition, $x' in mathscr L_v cap C_h$ if and only if
there exists a time $t in [0,infty)$ such that $x' = x + tv$ and $h(x') = 0$.
Thus,
begin{eqnarray}
  r_h(x) = inf{t mid (v,t) in mathbb S_{m-1} times [0,infty),; h(x + tv) = 0}.
  tag{1}
end{eqnarray}
The convex case
If $h$ is convex and differentiable, then for all $(v,t) in mathbb S_{m-1} times
[0,infty)$ one has
begin{eqnarray*}
  begin{split}
    h(x+tv) &ge h(x) + tnabla h(x)^Tvge h(x) - t|nabla h(x)|,\
    -h(x+tv) &le -h(x) - tnabla h(x)^Tv le -h(x) + t|nabla h(x)|,    
end{split}
end{eqnarray*}
where we have used Cauchy-Schwarz to reach the final term of each chain of
inequalities. Combining the above inequalities and (1) one
computes
begin{eqnarray*}
  begin{split}
r_h(x) &ge inf{t ge 0 mid h(x) -
t|nabla h(x)|le 0} = frac{max(h(x),0)}{|nabla h(x)|},\
r_h(x) &ge inf{t ge 0 mid -h(x) +
t|nabla h(x)|ge 0} = frac{max(-h(x), 0)}{|nabla h(x)|},
end{split}
end{eqnarray*}
We have thus proved

Theorem.
If $h:mathbb R^m to mathbb R$ is convex differentiable, we have
begin{eqnarray}
  r_h(x) ge frac{|h(x)|}{|nabla h(x)|},;forall x in mathbb R^m
end{eqnarray}

Lipschitz case
If we instead assume $h$ Lipschitz continuous (but not necessarily convex), then $|h(x + tv) -  h(x)| le
Lt$ for all $(v,t) in S_{m-1} times [0,infty)$. Using (1), we may then compute
begin{eqnarray}
  begin{split}
r_h(x) &ge inf{t ge 0mid h(x) - tL ge 0} =
frac{max(h(x),0)}{L},\
r_h(x) &ge inf{t ge 0mid h(x) + tL ge 0} =
frac{max(-h(x),0)}{L}.
end{split}
end{eqnarray}
Putting things together, we get

Theorem. If $h:mathbb R^m to mathbb R$ is $L$-Lipschitz, then we have
begin{eqnarray}
  r_h(x) ge frac{|h(x)|}{L},;forall x in mathbb R^m.
end{eqnarray}

Given convex differentiable function $h: R^m to R$, what is geometric interpretation of $h(x)/|nabla h(x)|$ vis à vis the level-set $h=0$?

One Answer

The convex case

Lipschitz case

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