Geometry Lesson Plan

Description

• cours - matière potentielle : plan
• cours - matière : geometry

Geometry Lesson Plan Curriculum/Instructional Strategies: This is a geometry lesson plan. Students will calculate the volume and the surface area of some specific objects. Teacher Direct Instruction (TDI) will be used in teaching this lesson. First, students will be provided the objective of the day. If there are multiple objectives, students will be introduced to them one by one. One objective will be fully taught then the class will be moved to the next one.

• arrow point out the height of the following shape
• shape on the screen
• meaning of volume
• rectangular cube
• area of the following circle
• height
• surface area
• surface- area
• cylinder
• lesson

Sujets

• Geometry
• Microsoft
• Meaning
• Surface
• Screen
• Following
• Rectángulo
• Cylinder
• Height
• Lesson

Informations

• Publié par: mochug
• Nombre de lectures: 18
• Langue: English

Extrait

Mathematical Research Letters xxx, 10001{100NN (2008)
FINITE BOUNDS FOR HOLDER-BRASCAMP-LIEB
MULTILINEAR INEQUALITIES
Jonathan Bennett, Anthony Carbery, Michael Christ,
and Terence Tao
Abstract. A criterion is established for the validity of multilinear inequalities of
a class considered by Brascamp and Lieb, generalizing well-known inequalties of
Rogers and H older, Young, and Loomis-Whitney.
1. Formulation
Consider multilinear functionals
Z mY
(1.1) ( f ;f ; ;f ) = f (‘ (y))dy1 2 m j j
nR j=1
n n nj jwhere each ‘ :R !R is a surjective linear transformation, and f :R !j j
[0; +1]. Let p ; ;p 2 [1;1]. For which m-tuples of exponents and linear1 m
transformations is
( f ;f ; ;f )1 2 m
Q(1.2) sup <1?
pkfk jf ;;f j L1 m j
The supremum is taken over all m-tuples of nonnegative Lebesgue measurable
functionsf having positive, nite norms. If n =n for every indexj then (1.2)j j
1is essentially a restatement of H older’s inequality. Other well-known particu-
lar cases include Young’s inequality for convolutions and the Loomis-Whitney
2inequality [15].
In this paper we characterize niteness of the supremum (1.2) in linear alge-
braic terms, and discuss certain variants and a generalization. The problem has
a long history, including the early work of Rogers [17] and H older [12]. In this
level of generality, the question was to our knowledge rst posed by Brascamp
and Lieb [4]. A primitive version of the problem involving Cartesian product
rather than linear algebraic structure was posed and solved by Finner [10]; see
x7 below. In the case when the dimension n of each target space equals one,j
Barthe [1] characterized (1.2). Carlen, Lieb and Loss [7] gave an alternative
Received by the editors October 14, 2008.
The third author was supported in part by NSF grant DMS-040126.
1For a discussion of the history of H older’s inequality, including its discovery by Rogers
[17], see [16].
2Loomis and Whitney considered only the special case where each f is the characteristicj
function of a set.
1000110002 Jonathan Bennett, Anthony Carbery, Michael Christ, and Terence Tao
characterization, closely related to ours, and an alternative proof for that case.
[7] developed an inductive analysis closely related to that of Finner, whose ar-
gument in turn relied on a slicing and induction argument employed earlier by
Loomis and Whitney [15] and Calder on [6] to treat special cases. [7] also in-
troduced a version of the key concepts of critical and subcritical subspaces, a
higher-dimensional reformulation of which is essential in our work.
An alternative line of analysis exists. Although rearrangement inequalities
such as that of Brascamp, Lieb, and Luttinger [5] do not apply when the target
spaces have dimensions greater than one, Lieb [14] nonetheless showed that the
supremum in (1.2) equals the supremum over all m-tuples of Gaussian func-
3tions, meaning those of the formf = exp( Q (y;y)) for some positive de nitej j
quadratic formQ . See [7] and references cited there for more on this approach.j
In a companion paper [3] we have given other proofs of our characterization of
(1.2), by using heat ow to continuously deform arbitrary functions f to Gaus-j
sians while increasing the ratio in (1.2). That approach extends work of Carlen,
Lieb, and Loss [7] via a method which they introduced.
We are indebted to a referee, whose careful reading and comments have im-
proved the exposition.
2. Results
Denote by dim (V ) the dimension of a vector space V , and by codim (V )W
the codimension of a subspace V W in W . It is convenient to reformulate
the problem in a more invariant fashion. Let H;H ;:::;H be Hilbert spaces1 m
of nite, positive dimensions. Each is equipped with a canonical Lebesgue mea-
dim (H)sure, by choosing orthonormal bases, thus obtaining identi cations with R ,
dim (H )jR . Let ‘ :H!H be surjective linear mappings. Let f :H !R bej j j jR Qm
nonnegative. Then ( f ; ;f ) equals f ‘ (y)dy.1 m j jH j=1
Theorem 2.1. For 1jm let H;H be Hilbert spaces of nite, positive di-j
mensions. For each indexj let‘ :H!H be surjective linear transformations,j j
and let p 2 [1;1]. Then (1.2) holds if and only ifj
X
1(2.1) dim (H) = p dim (H )jj
j
and
X
1(2.2) dim (V ) p dim (‘ (V )) for every subspace V H:jj
j
This equivalence is established by other methods in [3], Theorem 1.15.
3This situation should be contrasted with that of multilinear operators of the same general
p qjform, mapping
L toL . Whenq 1, such multilinear operators are equivalent by dualityj
to multilinear forms . This is not so for q < 1, and Gaussians are then quite far from being
extremal [8].HOLDER-BRASCAMP-LIEB MULTILINEAR INEQUALITIES 10003
Given that (2.1) holds, the hypothesis (2.2) can be equivalently restated as
X
1
(2.3) codim (V ) p codim (‘ (V )) for every subspace V H;H H jj j
j
any two of these three conditions (2.1), (2.2), (2.3) imply the third. As will
be seen through the discussion of variants below, (2.2) expresses a necessary
condition governing large-scale geometry (compare Theorem 2.5), while (2.3)
expresses a necessary condition governing small-scale geometry (compare Theo-
rem 2.2). See also the discussion of necessary conditions for Theorem 2.3.
In the rank one case, when each target space H is one-dimensional, a nec-j
essary and su cient condition for inequality (1.2) was rst obtained by Barthe
[1]. Carlen, Lieb, and Loss [7] gave a di erent proof of the inequality for the
rank one case, and a di erent characterization which is closely related to ours.
Write‘ (x) =hx;vi. It was shown in [7] that (1.2) is equivalent, in the rank onej jP P
1 1case, to having p = dim (H) and p dim (span (fv : j2 Sg))jj j j2S j
for every subset S off1; 2; ;mg; a set of indices S was said to be subcritical
if this last inequality holds, and to be critical if it holds with equality. In the
higher-rank case, we have formulated these concepts as properties of subspaces
of H, rather than of subsets off1; 2; ;mg.
To elucidate the connection between the two formulations in the rank one
case, dene W = spanfv :j2Sg, and say that a set of indices S is maximalS j
~if there is no larger set S of indices satisfying W =W . All sets of indices are~ SS
subcritical, if and only if all maximal sets of indices are subcritical. Ifj2S then
? ?codim (‘ (W )) = 1; if j2= S and S is maximal then codim (‘ (W )) = 0;H j H jj jS S
?and codim(W ) = dim (span (fv : j2 Sg)). Thus if S is maximal, then thejS Pn 1 ? ?subcriticality of S is equivalent to p codim (‘ (W )) codim(W ).H jj=1 j j S SPn 1As noted above, under the condition p = dim (H), this is equivalentj=1 jP 1
to our subcriticality condition dim (V ) p dim (‘ (V )) for the subspacejjj
?V =W .S
The necessity of (2.1) follows from scaling: if f (x ) =g ( x ) for each 2j j jj Q
+ dim (H) R then ( ff g) is proportional to , while kf k is proportionalpj j jjQ
dim (H )=pj jto . That (2.2) is also necessary will be shown inx5 in thej
course of the proof of the more general Theorem 2.3.
Remark 2.1. can be alternatively expressed as a constant multiple of theR Q
integral f d , where is a linear subspace of H and is Lebesguej j j j
measure on . More exactly, is the range of the map H 3 x7! ‘ (x).j j
Denote by the restriction to of the natural projection : H ! H .j j i i j
Then condition (2.2) can be restated as
X
1~ ~ ~(2.4) dim ( ) p dim ( ( )) for every linear subspace .jj
j10004 Jonathan Bennett, Anthony Carbery, Michael Christ, and Terence Tao
A local variant is also natural. Consider
Z Y
(2.5) (f ; ;f ) = f ‘ (y)dy:loc 1 m j j
fy2H:jyj 1g j
Theorem 2.2. Let H;H ;‘ , and f :H ! [0;1) be as in Theorem 2.1. Letj j j j
p 2 [1;1] for 1 j m. A necessary and su cient condition for there toj
exist C <1 such that
Y
p(2.6) (f ; ;f )C kfk jloc 1 m L
j
for all nonnegative measurable functionsf is that every subspaceV ofH satis esjP 1(2.3): codim (V ) p codim (‘ (V )).H H jj j j
This is equivalent to Theorem 8.17 of [3], proved there by a di erent method.
Certain cases of 2.2 follow from Theorem 2.1; if there exist exponents
r satisfying the hypotheses (2.1) and (2.2) of Theorem 2.1, such that r j j
p for all j, then the conclusion of Theorem 2.2 follows directly from that ofj
0Theorem 2.1 by H older’s inequality, since kfk r C kfk p . But not allj j j jL L
cases of Theorem 2.2 are subsumed in Theorem 2.1 in this way. See Remark 7.1
for examples.
The next theorem, in which some but not necessarily all coordinates of y are
constrained to a bounded set, uni es Theorems 2.1 and 2.2.
Theorem 2.3. Let H;H ; ;H be nite-dimensional Hilbert spaces and as-0 m
sume that dim (H )> 0 for allj 1. Let‘ :H!H be linear transformationsj j j
for 0jm, which are surjective for all j 1. Let p 2 [1;1] for 1jm.j
Then there exists C <1 such that
Z m mY Y
p(2.7) f ‘ (y)dyC kfk jj j j L
fy2H:j‘ (y)j 1g0 j=1 j=1
for all nonnegative Lebesgue measurable functions f if and only ifj
mX
1(2.8) dim (V ) p dim (‘ (V )) for all subspaces V kernel (‘ )j 0j
j=1
and
mX
1
(2.9) codim (V ) p codim (‘ (V )) for all subspaces V H.H H jj j
j=1
This subsumes Theorem 2.2, by taking H = H and ‘ : H ! H to be0 0
the identity; (2.8) then only applies tof0g, for which it holds automatically,
so