For a convex body K⊂ℝd we investigate three associated bodies, its intersection body IK (for 0∈in... more For a convex body K⊂ℝd we investigate three associated bodies, its intersection body IK (for 0∈int K), cross-section body CK, and projection body IIK, which satisfy IK⊂CK⊂IIK. Conversely we prove CK⊂const1(d)I(K−x) for some x∈int K, and IIK⊂const2 (d)CK, for certain constants, the first constant being sharp. We estimate the maximal k-volume of sections of 1/2(K+(-K)) with k-planes parallel to a fixed k-plane by the analogous quantity for K; our inequality is, if only k is fixed, sharp. For L⊂ℝd a convex body, we take n random segments in L, and consider their ‘Minkowski average’ D. We prove that, for V(L) fixed, the supremum of V(D) (with also n∈N arbitrary) is minimal for L an ellipsoid. This result implies the Petty projection inequality about max V((IIM)*), for M⊂ℝd a convex body, with V(M) fixed. We compare the volumes of projections of convex bodies and the volumes of the projections of their sections, and, dually, the volumes of sections of convex bodies and the volumes of sections of their circumscribed cylinders. For fixed n, the pth moments of V(D) (1≤pV(L) fixed, by the ellipsoids. For k=2, the supremum (n∈N arbitrary) and the pth moment (n fixed) of V(D) are maximized for example by triangles, and, for L centrally symmetric, for example by parallelograms. Last we discuss some examples for cross-section bodies.
The platelet-derived growth factor (PDGF) family consists of three different dimeric forms, AA, B... more The platelet-derived growth factor (PDGF) family consists of three different dimeric forms, AA, BB, and AB, of the two consitituent polypeptide chains, A and B. These interact with two different cell surface receptors that, in part, mediate different cellular functions. The various forms of PDGF, as well as the receptors, are expressed at high frequency in glioblastoma multiforme, and it has been suggested that the growth of this tumor might be affected by autocrine loops involving PDGF and its receptors. The present paper focuses on recent discoveries regarding the family of PDGF ligands and receptors, as well as reviews results concerning PDGF-dependent autocrine growth in experimental and spontaneous glioblastoma.
Tumor cells may stimulate their own proliferation through an autocrine mechanism by simultaneousl... more Tumor cells may stimulate their own proliferation through an autocrine mechanism by simultaneously producing growth factors and growth factor receptors. We now report that numerous human tumor-derived cell lines simultaneously express the genes for platelet-derived growth factor (PDGF) A and B chains and the PDGF receptor (PDGF-R). Measurement of mRNA transcribed from these genes showed that among 16 malignant glioma cell lines tested, 15 expressed the PDGF A gene, 12 expressed the PDGF B gene, and 13 expressed the PDGF-R gene. Of three osteosarcoma lines, three expressed PDGF A, two expressed PDGF B, and three expressed PDGF-R. For eight malignant melanoma lines, seven expressed PDGF A, five expressed PDGF B, and three expressed PDGF-R genes. Thus, 13 of 16 malignant glioma, 3 of 3 osteosarcomas, and 3 of 8 malignant melanoma cell lines expressed the PDGF receptor gene and either or both PDGF genes. Five cell lines were tested for production of biologically active PDGF and PDGF receptor protein. Media conditioned by each of the five cell lines induced tyrosine phosphorylation of a protein identical in size to the PDGF receptor. These five cell lines also produced PDGF receptor protein as measured by Western blot analysis or metabolic labeling and immunoprecipitation using PDGF-R antibodies. The PDGF receptors of these cell lines were activated by human platelet PDGF or by recombinant AA or BB homodimers. Intracellular interaction of these receptors with the growth factor simultaneously produced may provide continuous stimulation to the proliferation of these cells.
For a convex body K⊂ℝd we investigate three associated bodies, its intersection body IK (for 0∈in... more For a convex body K⊂ℝd we investigate three associated bodies, its intersection body IK (for 0∈int K), cross-section body CK, and projection body IIK, which satisfy IK⊂CK⊂IIK. Conversely we prove CK⊂const1(d)I(K−x) for some x∈int K, and IIK⊂const2 (d)CK, for certain constants, the first constant being sharp. We estimate the maximal k-volume of sections of 1/2(K+(-K)) with k-planes parallel to a fixed k-plane by the analogous quantity for K; our inequality is, if only k is fixed, sharp. For L⊂ℝd a convex body, we take n random segments in L, and consider their ‘Minkowski average’ D. We prove that, for V(L) fixed, the supremum of V(D) (with also n∈N arbitrary) is minimal for L an ellipsoid. This result implies the Petty projection inequality about max V((IIM)*), for M⊂ℝd a convex body, with V(M) fixed. We compare the volumes of projections of convex bodies and the volumes of the projections of their sections, and, dually, the volumes of sections of convex bodies and the volumes of sections of their circumscribed cylinders. For fixed n, the pth moments of V(D) (1≤pV(L) fixed, by the ellipsoids. For k=2, the supremum (n∈N arbitrary) and the pth moment (n fixed) of V(D) are maximized for example by triangles, and, for L centrally symmetric, for example by parallelograms. Last we discuss some examples for cross-section bodies.
The platelet-derived growth factor (PDGF) family consists of three different dimeric forms, AA, B... more The platelet-derived growth factor (PDGF) family consists of three different dimeric forms, AA, BB, and AB, of the two consitituent polypeptide chains, A and B. These interact with two different cell surface receptors that, in part, mediate different cellular functions. The various forms of PDGF, as well as the receptors, are expressed at high frequency in glioblastoma multiforme, and it has been suggested that the growth of this tumor might be affected by autocrine loops involving PDGF and its receptors. The present paper focuses on recent discoveries regarding the family of PDGF ligands and receptors, as well as reviews results concerning PDGF-dependent autocrine growth in experimental and spontaneous glioblastoma.
Tumor cells may stimulate their own proliferation through an autocrine mechanism by simultaneousl... more Tumor cells may stimulate their own proliferation through an autocrine mechanism by simultaneously producing growth factors and growth factor receptors. We now report that numerous human tumor-derived cell lines simultaneously express the genes for platelet-derived growth factor (PDGF) A and B chains and the PDGF receptor (PDGF-R). Measurement of mRNA transcribed from these genes showed that among 16 malignant glioma cell lines tested, 15 expressed the PDGF A gene, 12 expressed the PDGF B gene, and 13 expressed the PDGF-R gene. Of three osteosarcoma lines, three expressed PDGF A, two expressed PDGF B, and three expressed PDGF-R. For eight malignant melanoma lines, seven expressed PDGF A, five expressed PDGF B, and three expressed PDGF-R genes. Thus, 13 of 16 malignant glioma, 3 of 3 osteosarcomas, and 3 of 8 malignant melanoma cell lines expressed the PDGF receptor gene and either or both PDGF genes. Five cell lines were tested for production of biologically active PDGF and PDGF receptor protein. Media conditioned by each of the five cell lines induced tyrosine phosphorylation of a protein identical in size to the PDGF receptor. These five cell lines also produced PDGF receptor protein as measured by Western blot analysis or metabolic labeling and immunoprecipitation using PDGF-R antibodies. The PDGF receptors of these cell lines were activated by human platelet PDGF or by recombinant AA or BB homodimers. Intracellular interaction of these receptors with the growth factor simultaneously produced may provide continuous stimulation to the proliferation of these cells.
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