Symmetry of the second derivative #
We show that, over the reals, the second derivative is symmetric.
The most precise result is Convex.second_derivative_within_at_symmetric
. It asserts that,
if a function is differentiable inside a convex set s
with nonempty interior, and has a second
derivative within s
at a point x
, then this second derivative at x
is symmetric. Note that
this result does not require continuity of the first derivative.
The following particular cases of this statement are especially relevant:
second_derivative_symmetric_of_eventually
asserts that, if a function is differentiable on a
neighborhood of x
, and has a second derivative at x
, then this second derivative is symmetric.
second_derivative_symmetric
asserts that, if a function is differentiable, and has a second
derivative at x
, then this second derivative is symmetric.
Implementation note #
For the proof, we obtain an asymptotic expansion to order two of f (x + v + w) - f (x + v)
, by
using the mean value inequality applied to a suitable function along the
segment [x + v, x + v + w]
. This expansion involves f'' ⬝ w
as we move along a segment directed
by w
(see Convex.taylor_approx_two_segment
).
Consider the alternate sum f (x + v + w) + f x - f (x + v) - f (x + w)
, corresponding to the
values of f
along a rectangle based at x
with sides v
and w
. One can write it using the two
sides directed by w
, as (f (x + v + w) - f (x + v)) - (f (x + w) - f x)
. Together with the
previous asymptotic expansion, one deduces that it equals f'' v w + o(1)
when v, w
tends to 0
.
Exchanging the roles of v
and w
, one instead gets an asymptotic expansion f'' w v
, from which
the equality f'' v w = f'' w v
follows.
In our most general statement, we only assume that f
is differentiable inside a convex set s
, so
a few modifications have to be made. Since we don't assume continuity of f
at x
, we consider
instead the rectangle based at x + v + w
with sides v
and w
,
in Convex.isLittleO_alternate_sum_square
, but the argument is essentially the same. It only works
when v
and w
both point towards the interior of s
, to make sure that all the sides of the
rectangle are contained in s
by convexity. The general case follows by linearity, though.
Assume that f
is differentiable inside a convex set s
, and that its derivative f'
is
differentiable at a point x
. Then, given two vectors v
and w
pointing inside s
, one can
Taylor-expand to order two the function f
on the segment [x + h v, x + h (v + w)]
, giving a
bilinear estimate for f (x + hv + hw) - f (x + hv)
in terms of f' w
and of f'' ⬝ w
, up to
o(h^2)
.
This is a technical statement used to show that the second derivative is symmetric.
One can get f'' v w
as the limit of h ^ (-2)
times the alternate sum of the values of f
along the vertices of a quadrilateral with sides h v
and h w
based at x
.
In a setting where f
is not guaranteed to be continuous at f
, we can still
get this if we use a quadrilateral based at h v + h w
.
Assume that f
is differentiable inside a convex set s
, and that its derivative f'
is
differentiable at a point x
. Then, given two vectors v
and w
pointing inside s
, one
has f'' v w = f'' w v
. Superseded by Convex.second_derivative_within_at_symmetric
, which
removes the assumption that v
and w
point inside s
.
If a function is differentiable inside a convex set with nonempty interior, and has a second derivative at a point of this convex set, then this second derivative is symmetric.
If a function is differentiable around x
, and has two derivatives at x
, then the second
derivative is symmetric.
If a function is differentiable, and has two derivatives at x
, then the second
derivative is symmetric.