differentiability of convex functions
in [Math]
|
Prev: CONTINUUM SUM DEBUNKED??? :(
Next: Exactly why the theories of relativity are complete nonsense- the basic mistake exposed!
From: Yihong on 7 Jan 2010 03:04 > Not even that. The best you can say in R^d is that f > is differentiable > a.e. True. I edited my previous post after I checked Rockafellar's book (Thm 25.5), but I guess it is too late to update. > Sets of measure 0 aren't necessarily countable. > And, IIRC, every > set of measure 0 is the set of non-differentiability > of some convex > function. > > -- Ron Bruck For R, yes, as constructed for instance in this paper linkinghub.elsevier.com/retrieve/pii/S0096300302009323 Not sure about general subset of zero Lebesgue measure.
From: Robert Israel on 7 Jan 2010 16:29 On Thu, 7 Jan 2010 10:09:00 -0800 (PST), Dave L. Renfro wrote: > Ronald Buck wrote: > >> Not even that. The best you can say in R^d is that >> f is differentiable a.e. Sets of measure 0 aren't >> necessarily countable. And, IIRC, every set of >> measure 0 is the set of non-differentiability of >> some convex function. > > This isn't true (and probably isn't what you intended to say), > since the set of non-differentiability of an arbitrary function > is G_delta_sigma (and without knowing this, it would still > be enough to know the non-differentiability set of a continuous > function is a Borel set). What might be true (and what you > probably meant) is that given any set E of measure 0, there > exists a convex function f whose set of non-differentiability > is a superset of E (i.e. f does not have a derivative at > each point in E). > > Dave L. Renfro Not even that. A convex function is (Gateaux) differentiable on a dense G_delta (Mazur's theorem). The OP was talking about Frechet differentiable, but let's stick to functions on R so these are the same. Let E be a dense G_delta of measure 0, and there will be no convex function f whose set of non-differentiability includes E. -- Robert Israel israel(a)math.MyUniversitysInitials.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada
From: Yihong on 7 Jan 2010 06:57 > > Let f: R^n -> R be defined as f(x) = max{<a, x>: a > > \in A}, where A is a > > compact subset of R^n. Then f is a convex function > > hence differentiable (I > > mean total differentiable i.e. > > Frechet-differentiable) everywhere but a > > countable number of points. > > That theorem is for a function with real domain, not > R^n. In R^n it > could fail differentiability on a curve, for example. > > > > > I wonder if there is any sufficient condition on A > > that guarantees the > > differentiability of f everywhere? Thanks! > > A non-differentiable (at a point) example: > n=1, A = {1,-1}, so f(x) = |x|. > > Now, before asking for a general condition to get f > differentiable > everywhere, find some interesting examples in R^1 > where it occurs. My guess is that A being a convex set with smooth boundary. Not sure how to it follows rigorously.
From: Robert Israel on 8 Jan 2010 02:03 On Thu, 07 Jan 2010 16:57:29 EST, Yihong wrote: >>> Let f: R^n -> R be defined as f(x) = max{<a, x>: a >>> \in A}, where A is a >>> compact subset of R^n. This is called the support function of A. See <http://eom.springer.de/S/s091270.htm>. >>> Then f is a convex function >>> hence differentiable (I >>> mean total differentiable i.e. >>> Frechet-differentiable) everywhere but a >>> countable number of points. >> >> That theorem is for a function with real domain, not >> R^n. In R^n it >> could fail differentiability on a curve, for example. >> >>> >>> I wonder if there is any sufficient condition on A >>> that guarantees the >>> differentiability of f everywhere? Thanks! >> >> A non-differentiable (at a point) example: >> n=1, A = {1,-1}, so f(x) = |x|. >> >> Now, before asking for a general condition to get f >> differentiable >> everywhere, find some interesting examples in R^1 >> where it occurs. > > My guess is that A being a convex set with smooth boundary. Not sure how to it follows rigorously. f is non-differentiable at the origin, except in trivial cases. IIRC it will be differentiable elsewhere if A is strictly convex. But straight line segments in the boundary will cause non-differentiability at points corresponding to supporting hyperplanes containing such segments. -- Robert Israel israel(a)math.MyUniversitysInitials.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada
From: Yihong on 13 Jan 2010 14:01
> On Thu, 07 Jan 2010 16:57:29 EST, Yihong wrote: > > >>> Let f: R^n -> R be defined as f(x) = max{<a, x>: > >>> a > >>> \in A}, where A is a > >>> compact subset of R^n. > > This is called the support function of A. See > <http://eom.springer.de/S/s091270.htm>. > > > >>> Then f is a convex function > >>> hence differentiable (I > >>> mean total differentiable i.e. > >>> Frechet-differentiable) everywhere but a > >>> countable number of points. > >> > >> That theorem is for a function with real domain, > >> not > >> R^n. In R^n it > >> could fail differentiability on a curve, for > >> example. > >> > >>> > >>> I wonder if there is any sufficient condition on > >>> A > >>> that guarantees the > >>> differentiability of f everywhere? Thanks! > >> > >> A non-differentiable (at a point) example: > >> n=1, A = {1,-1}, so f(x) = |x|. > >> > >> Now, before asking for a general condition to get > >> f > >> differentiable > >> everywhere, find some interesting examples in R^1 > >> where it occurs. > > > > My guess is that A being a convex set with smooth > > boundary. Not sure how to it follows rigorously. > > f is non-differentiable at the origin, except in > trivial cases. > IIRC it will be differentiable elsewhere if A is > strictly convex. > But straight line segments in the boundary will cause > non-differentiability > at points corresponding to supporting hyperplanes > containing such segments. > > -- > Robert Israel > israel(a)math.MyUniversitysInitials.ca > Department of Mathematics > http://www.math.ubc.ca/~israel > University of British Columbia Vancouver, > BC, Canada Thanks a lot! Could you please give me a reference about the differentiability when A is strictly convex (I presume by strict convexity you mean p x + (1-p) y \in int(A) for all 0<p<1 and all x,y \in A). I was told before that similar stuff about support function can be found in the book by Hiriart-Urruty and Lemarechal, but no luck yet. Yihong |